{"id":68,"date":"2017-11-14T13:47:10","date_gmt":"2017-11-14T13:47:10","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/chapter\/scalars-and-vectors\/"},"modified":"2017-11-18T01:29:04","modified_gmt":"2017-11-18T01:29:04","slug":"scalars-and-vectors","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/chapter\/scalars-and-vectors\/","title":{"raw":"Scalars and Vectors","rendered":"Scalars and Vectors"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Describe the difference between vector and scalar quantities.<\/li>\n<li>Identify the magnitude and direction of a vector.<\/li>\n<li>Explain the effect of multiplying a vector quantity by a scalar.<\/li>\n<li>Describe how one-dimensional vector quantities are added or subtracted.<\/li>\n<li>Explain the geometric construction for the addition or subtraction of vectors in a plane.<\/li>\n<li>Distinguish between a vector equation and a scalar equation.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167132379492\">Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, \u201ca class period lasts 50 min\u201d or \u201cthe gas tank in my car holds 65 L\u201d or \u201cthe distance between two posts is 100 m.\u201d A physical quantity that can be specified completely in this manner is called a <span>scalar quantity<\/span>. Scalar is a synonym of \u201cnumber.\u201d Time, mass, distance, length, volume, temperature, and energy are examples of <span>scalar<\/span> quantities.<\/p>\n<p id=\"fs-id1167133331747\">Scalar quantities that have the same physical units can be added or subtracted according to the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50 min lasts [latex]50\\phantom{\\rule{0.2em}{0ex}}\\text{min}-10\\phantom{\\rule{0.2em}{0ex}}\\text{min}=40\\phantom{\\rule{0.2em}{0ex}}\\text{min}[\/latex]. Similarly, a 60-cal serving of corn followed by a 200-cal serving of donuts gives [latex]60\\phantom{\\rule{0.2em}{0ex}}\\text{cal}+200\\phantom{\\rule{0.2em}{0ex}}\\text{cal}=260\\phantom{\\rule{0.2em}{0ex}}\\text{cal}[\/latex] of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. For example, if yesterday\u2019s breakfast had 200 cal of energy and today\u2019s breakfast has four times as much energy as it had yesterday, then today\u2019s breakfast has [latex]4\\left(200\\phantom{\\rule{0.2em}{0ex}}\\text{cal}\\right)=800\\phantom{\\rule{0.2em}{0ex}}\\text{cal}[\/latex] of energy. Two scalar quantities can also be multiplied or divided by each other to form a derived scalar quantity. For example, if a train covers a distance of 100 km in 1.0 h, its speed is 100.0 km\/1.0 h = 27.8 m\/s, where the speed is a derived scalar quantity obtained by dividing distance by time.<\/p>\n<p id=\"fs-id1167132683784\">Many physical quantities, however, cannot be described completely by just a single number of physical units. For example, when the U.S. Coast Guard dispatches a ship or a helicopter for a rescue mission, the rescue team must know not only the distance to the distress signal, but also the direction from which the signal is coming so they can get to its origin as quickly as possible. Physical quantities specified completely by giving a number of units (magnitude) and a direction are called <span>vector quantities<\/span>. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called <span>vectors<\/span> (<a href=\"#CNX_UPhysics_02_01_dog\" class=\"autogenerated-content\">(Figure)<\/a>). We can add or subtract two vectors, and we can multiply a vector by a scalar or by another vector, but we cannot divide by a vector. The operation of division by a vector is not defined.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_dog\">\n<div class=\"bc-figcaption figcaption\">We draw a vector from the initial point or origin (called the \u201ctail\u201d of a vector) to the end or terminal point (called the \u201chead\u201d of a vector), marked by an arrowhead. Magnitude is the length of a vector and is always a positive scalar quantity. (credit: modification of work by Cate Sevilla)<\/div>\n<p><span id=\"fs-id1167133854996\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_dog.jpg\" alt=\"A photo of a dog. Below the photo is a horizontal arrow which starts below the dog\u2019s tail and ends below the dog\u2019s nose. The arrow is labeled Vector D, and its length is labeled as magnitude D. The start (tail) of the arrow is labeled \u201cFrom rail of a vector origin\u201d and its end (head) is labeled \u201cTo head of a vector end.\u201d\"><\/span><\/p><\/div>\n<p id=\"fs-id1167133527625\">Let\u2019s examine vector algebra using a graphical method to be aware of basic terms and to develop a qualitative understanding. In practice, however, when it comes to solving physics problems, we use analytical methods, which we\u2019ll see in the next section. Analytical methods are more simple computationally and more accurate than graphical methods. From now on, to distinguish between a vector and a scalar quantity, we adopt the common convention that a letter in bold type with an arrow above it denotes a vector, and a letter without an arrow denotes a scalar. For example, a distance of 2.0 km, which is a scalar quantity, is denoted by <em>d<\/em> = 2.0 km, whereas a displacement of 2.0 km in some direction, which is a vector quantity, is denoted by [latex]\\stackrel{\\to }{d}[\/latex].<\/p>\n<p id=\"fs-id1167133333619\">Suppose you tell a friend on a camping trip that you have discovered a terrific fishing hole 6 km from your tent. It is unlikely your friend would be able to find the hole easily unless you also communicate the direction in which it can be found with respect to your campsite. You may say, for example, \u201cWalk about 6 km northeast from my tent.\u201d The key concept here is that you have to give not one but <em>two<\/em> pieces of information\u2014namely, the distance or magnitude (6 km) <em>and<\/em> the direction (northeast).<\/p>\n<p id=\"fs-id1167133802941\">Displacement is a general term used to describe a <em>change in position<\/em>, such as during a trip from the tent to the fishing hole. Displacement is an example of a vector quantity. If you walk from the tent (location <em>A<\/em>) to the hole (location <em>B<\/em>), as shown in <a href=\"#CNX_UPhysics_02_01_hole\" class=\"autogenerated-content\">(Figure)<\/a>, the vector [latex]\\stackrel{\\to }{D}[\/latex], representing your <span>displacement<\/span>, is drawn as the arrow that originates at point <em>A<\/em> and ends at point <em>B<\/em>. The arrowhead marks the end of the vector. The direction of the displacement vector [latex]\\stackrel{\\to }{D}[\/latex] is the direction of the arrow. The length of the arrow represents the <span>magnitude<\/span> <em>D<\/em> of vector [latex]\\stackrel{\\to }{D}[\/latex]. Here, <em>D<\/em> = 6 km. Since the magnitude of a vector is its length, which is a positive number, the magnitude is also indicated by placing the absolute value notation around the symbol that denotes the vector; so, we can write equivalently that [latex]D\\equiv |\\stackrel{\\to }{D}|[\/latex]. To solve a vector problem graphically, we need to draw the vector [latex]\\stackrel{\\to }{D}[\/latex] to scale. For example, if we assume 1 unit of distance (1 km) is represented in the drawing by a line segment of length <em>u<\/em> = 2 cm, then the total displacement in this example is represented by a vector of length [latex]d=6u=6\\left(2\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)=12\\phantom{\\rule{0.2em}{0ex}}\\text{cm}[\/latex], as shown in <a href=\"#CNX_UPhysics_02_01_vector01\" class=\"autogenerated-content\">(Figure)<\/a>. Notice that here, to avoid confusion, we used [latex]D=6\\phantom{\\rule{0.2em}{0ex}}\\text{km}[\/latex] to denote the magnitude of the actual displacement and <em>d<\/em> = 12 cm to denote the length of its representation in the drawing.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_hole\">\n<div class=\"bc-figcaption figcaption\">The displacement vector from point <em>A<\/em> (the initial position at the campsite) to point <em>B<\/em> (the final position at the fishing hole) is indicated by an arrow with origin at point <em>A<\/em> and end at point <em>B<\/em>. The displacement is the same for any of the actual paths (dashed curves) that may be taken between points <em>A<\/em> and <em>B<\/em>.<\/div>\n<p><span id=\"fs-id1167133464547\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_hole.jpg\" alt=\"An illustration of a lake, some distance northeast from a tent. North is up on the page, east to the right. The tent is labeled as location A, and the lake as location B. A straight arrow starts at A and ends at B. Three meandering paths, shown as dashed lines, also start at A and end at B.\"><\/span><\/p><\/div>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector01\">\n<div class=\"bc-figcaption figcaption\">A displacement [latex]\\stackrel{\\to }{D}[\/latex] of magnitude 6 km is drawn to scale as a vector of length 12 cm when the length of 2 cm represents 1 unit of displacement (which in this case is 1 km).<\/div>\n<p><span id=\"fs-id1167133318847\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector01.jpg\" alt=\"A ruler is shown, with distance measured in centimeters. A vector is shown as an arrow parallel to the ruler, extending from its end at 0 c m to 12 c m, and is labeled as vector D.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167133824259\">Suppose your friend walks from the campsite at <em>A<\/em> to the fishing pond at <em>B<\/em> and then walks back: from the fishing pond at <em>B<\/em> to the campsite at <em>A<\/em>. The magnitude of the displacement vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex] from <em>A<\/em> to <em>B<\/em> is the same as the magnitude of the displacement vector [latex]{\\stackrel{\\to }{D}}_{BA}[\/latex] from <em>B<\/em> to <em>A<\/em> (it equals 6 km in both cases), so we can write [latex]{D}_{AB}={D}_{BA}[\/latex]. However, vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex] is <em>not<\/em> equal to vector [latex]{\\stackrel{\\to }{D}}_{BA}[\/latex] because these two vectors have different directions: [latex]{\\stackrel{\\to }{D}}_{AB}\\ne {\\stackrel{\\to }{D}}_{BA}[\/latex]. In <a href=\"#CNX_UPhysics_02_01_hole\" class=\"autogenerated-content\">(Figure)<\/a>, vector [latex]{\\stackrel{\\to }{D}}_{BA}[\/latex] would be represented by a vector with an origin at point <em>B<\/em> and an end at point <em>A<\/em>, indicating vector [latex]{\\stackrel{\\to }{D}}_{BA}[\/latex] points to the southwest, which is exactly [latex]180\\text{\u00b0}[\/latex] opposite to the direction of vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex]. We say that vector [latex]{\\stackrel{\\to }{D}}_{BA}[\/latex] is <span>antiparallel<\/span> to vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex] and write [latex]{\\stackrel{\\to }{D}}_{AB}=\\text{\u2212}{\\stackrel{\\to }{D}}_{BA}[\/latex], where the minus sign indicates the antiparallel direction.<\/p>\n<p id=\"fs-id1167133579247\">Two vectors that have identical directions are said to be <span>parallel vectors<\/span>\u2014meaning, they are <em>parallel<\/em> to each other. Two parallel vectors [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}[\/latex] are equal, denoted by [latex]\\stackrel{\\to }{A}=\\stackrel{\\to }{B}[\/latex], if and only if they have equal magnitudes [latex]|\\stackrel{\\to }{A}|=|\\stackrel{\\to }{B}|[\/latex]. Two vectors with directions perpendicular to each other are said to be <span>orthogonal vectors<\/span>. These relations between vectors are illustrated in <a href=\"#CNX_UPhysics_02_01_vector02\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector02\">\n<div class=\"bc-figcaption figcaption\">Various relations between two vectors [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}[\/latex]. (a) [latex]\\stackrel{\\to }{A}\\ne \\stackrel{\\to }{B}[\/latex] because [latex]A\\ne B[\/latex]. (b) [latex]\\stackrel{\\to }{A}\\ne \\stackrel{\\to }{B}[\/latex] because they are not parallel and [latex]A\\ne B[\/latex]. (c) [latex]\\stackrel{\\to }{A}\\ne \\text{\u2212}\\stackrel{\\to }{A}[\/latex] because they have different directions (even though [latex]|\\stackrel{\\to }{A}|=|-\\stackrel{\\to }{A}|=A\\right)[\/latex]. (d) [latex]\\stackrel{\\to }{A}=\\stackrel{\\to }{B}[\/latex] because they are parallel <em>and<\/em> have identical magnitudes <em>A<\/em> = <em>B.<\/em> (e) [latex]\\stackrel{\\to }{A}\\ne \\stackrel{\\to }{B}[\/latex] because they have different directions (are not parallel); here, their directions differ by [latex]90\\text{\u00b0}[\/latex]\u2014meaning, they are orthogonal.<\/div>\n<p><span id=\"fs-id1167132332563\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector02.jpg\" alt=\"Figure a: Two examples of vector A parallel to vector B. In one, A and B are on the same line, one after the other, but A is longer than B. In the other, A and B are parallel to each other with their tails aligned, but A is shorter than B. Figure b: An example of vector A antiparallel to vector B. Vector A points to the left and is longer than vector B, which points to the right. The angle between them is 180 degrees. Figure c: An example of vector A antiparallel to minus vector A: A points to the right and \u2013A points to the left. Both are the same length. Figure d: Two examples of vector A equal to vector B: In one, A and B are on the same line, one after the other, and both are the same length. In the other, A and B are parallel to each other with their tails aligned, and both are the same length. Figure e: Two examples of vector A orthogonal to vector B: In one, A points down and B points to the right, meeting at a right angle, and both are the same length. In the other, points down and to the right and B points down and to the left, meeting A at a right angle. Both are the same length.\"><\/span><\/p><\/div>\n<div id=\"fs-id1167132576708\" class=\"check-understanding\">\n<div id=\"fs-id1167132485087\">\n<div id=\"fs-id1167132240816\">\n<p id=\"fs-id1167132539326\"><strong>Check Your Understanding<\/strong> Two motorboats named <em>Alice<\/em> and <em>Bob<\/em> are moving on a lake. Given the information about their velocity vectors in each of the following situations, indicate whether their velocity vectors are equal or otherwise. (a) <em>Alice<\/em> moves north at 6 knots and <em>Bob<\/em> moves west at 6 knots. (b) <em>Alice<\/em> moves west at 6 knots and <em>Bob<\/em> moves west at 3 knots. (c) <em>Alice<\/em> moves northeast at 6 knots and <em>Bob<\/em> moves south at 3 knots. (d) <em>Alice<\/em> moves northeast at 6 knots and <em>Bob<\/em> moves southwest at 6 knots. (e) <em>Alice<\/em> moves northeast at 2 knots and <em>Bob<\/em> moves closer to the shore northeast at 2 knots.<\/p>\n<\/div>\n<div id=\"fs-id1167132519498\">\n<p id=\"fs-id1167132595446\">a. not equal because they are orthogonal; b. not equal because they have different magnitudes; c. not equal because they have different magnitudes and directions; d. not equal because they are antiparallel; e. equal.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1167132305565\">\n<h3>Algebra of Vectors in One Dimension<\/h3>\n<p id=\"fs-id1167132274736\">Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. We can illustrate these vector concepts using an example of the fishing trip seen in <a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_FishTrip\">\n<div class=\"bc-figcaption figcaption\">Displacement vectors for a fishing trip. (a) Stopping to rest at point <em>C<\/em> while walking from camp (point <em>A<\/em>) to the pond (point <em>B<\/em>). (b) Going back for the dropped tackle box (point <em>D<\/em>). (c) Finishing up at the fishing pond.<\/div>\n<p><span id=\"fs-id1167133792347\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_FishTrip.jpg\" alt=\"Three illustrations of the same tent and lake northeast of the tent. North is up on the page. The tent location is point A, and the lake location is point B. A location between A and B, about 2\/3 of the way from A to B, is labeled as point C. In figure a, the vector from A to B is shown as a blue arrow, starting at A and ending at B, and labeled vector D sub A B. The vector from A to C is shown as a red arrow, starting at A and ending at C and labeled vector D sub A C. Three meandering paths are shown as dashed lines that start at A and end at B. Figure b adds the following to the illustration of figure a: Point D is added about half way between point A and B. The vector from A to D is shown as a purple arrow, starting at A and ending at D and labeled vector D sub A D. The vector from D to B is shown as an orange arrow, starting at D and ending at B and labeled vector D sub D B. Figure c adds a green arrow from point C to point D and is labeled vector D sub C D. Vector D sub C D points in the direction opposite to that of the other vectors, toward the tent rather than toward the lake.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167133328285\">Suppose your friend departs from point <em>A<\/em> (the campsite) and walks in the direction to point <em>B<\/em> (the fishing pond), but, along the way, stops to rest at some point <em>C<\/em> located three-quarters of the distance between <em>A<\/em> and <em>B<\/em>, beginning from point <em>A<\/em> (<a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>(a)). What is his displacement vector [latex]{\\stackrel{\\to }{D}}_{AC}[\/latex] when he reaches point <em>C<\/em>? We know that if he walks all the way to <em>B<\/em>, his displacement vector relative to <em>A<\/em> is [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex], which has magnitude [latex]{D}_{AB}=6\\phantom{\\rule{0.2em}{0ex}}\\text{km}[\/latex] and a direction of northeast. If he walks only a 0.75 fraction of the total distance, maintaining the northeasterly direction, at point <em>C<\/em> he must be [latex]0.75{D}_{AB}=4.5\\phantom{\\rule{0.2em}{0ex}}\\text{km}[\/latex] away from the campsite at <em>A<\/em>. So, his displacement vector at the rest point <em>C<\/em> has magnitude [latex]{D}_{AC}=4.5\\phantom{\\rule{0.2em}{0ex}}\\text{km}=0.75{D}_{AB}[\/latex] and is parallel to the displacement vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex]. All of this can be stated succinctly in the form of the following <span>vector equation<\/span>:<\/p>\n<div id=\"fs-id1167132497239\" class=\"unnumbered\">[latex]{\\stackrel{\\to }{D}}_{AC}=0.75{\\stackrel{\\to }{D}}_{AB}.[\/latex]<\/div>\n<p id=\"fs-id1167132624842\">In a vector equation, both sides of the equation are vectors. The previous equation is an example of a vector multiplied by a positive scalar (number) [latex]\\alpha =0.75[\/latex]. The result, [latex]{\\stackrel{\\to }{D}}_{AC}[\/latex], of such a multiplication is a new vector with a direction parallel to the direction of the original vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex].<\/p>\n<p id=\"fs-id1167133675760\">In general, when a vector [latex]\\stackrel{\\to }{A}[\/latex] is multiplied by a <em>positive<\/em> scalar [latex]\\alpha [\/latex], the result is a new vector [latex]\\stackrel{\\to }{B}[\/latex] that is <em>parallel<\/em> to [latex]\\stackrel{\\to }{A}[\/latex]:<\/p>\n<div id=\"fs-id1167132262787\" class=\"equation-callout\">\n<div id=\"fs-id1167133686173\">[latex]\\stackrel{\\to }{B}=\\alpha \\stackrel{\\to }{A}.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167133750753\">The magnitude [latex]|\\stackrel{\\to }{B}|[\/latex] of this new vector is obtained by multiplying the magnitude [latex]|\\stackrel{\\to }{A}|[\/latex] of the original vector, as expressed by the <span>scalar equation<\/span>:<\/p>\n<div id=\"fs-id1167132320492\" class=\"equation-callout\">\n<div id=\"fs-id1167133394048\">[latex]B=|\\alpha |A.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167132413817\">In a scalar equation, both sides of the equation are numbers. <a href=\"#fs-id1167133394048\" class=\"autogenerated-content\">(Figure)<\/a> is a scalar equation because the magnitudes of vectors are scalar quantities (and positive numbers). If the scalar [latex]\\alpha [\/latex] is <em>negative<\/em> in the vector equation <a href=\"#fs-id1167133686173\" class=\"autogenerated-content\">(Figure)<\/a>, then the magnitude [latex]|\\stackrel{\\to }{B}|[\/latex] of the new vector is still given by <a href=\"#fs-id1167133394048\" class=\"autogenerated-content\">(Figure)<\/a>, but the direction of the new vector [latex]\\stackrel{\\to }{B}[\/latex] is <em>antiparallel<\/em> to the direction of [latex]\\stackrel{\\to }{A}[\/latex]. These principles are illustrated in <a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>(a) by two examples where the length of vector [latex]\\stackrel{\\to }{A}[\/latex] is 1.5 units. When [latex]\\alpha =2[\/latex], the new vector [latex]\\stackrel{\\to }{B}=2\\stackrel{\\to }{A}[\/latex] has length [latex]B=2A=3.0\\phantom{\\rule{0.2em}{0ex}}\\text{units}[\/latex] (twice as long as the original vector) and is parallel to the original vector. When [latex]\\alpha =-2[\/latex], the new vector [latex]\\stackrel{\\to }{C}=-2\\stackrel{\\to }{A}[\/latex] has length [latex]C=|-2|A=3.0\\phantom{\\rule{0.2em}{0ex}}\\text{units}[\/latex] (twice as long as the original vector) and is antiparallel to the original vector.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector03\">\n<div class=\"bc-figcaption figcaption\">Algebra of vectors in one dimension. (a) Multiplication by a scalar. (b) Addition of two vectors [latex]\\left(\\stackrel{\\to }{R}[\/latex] is called the <em>resultant<\/em> of vectors [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}\\right)[\/latex]. (c) Subtraction of two vectors [latex]\\left(\\stackrel{\\to }{D}[\/latex] is the difference of vectors [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}\\right)[\/latex].<\/div>\n<p><span id=\"fs-id1167132221972\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector03.jpg\" alt=\"Figure a shows vector A pointing to the right. It has magnitude A=1.5. Vector B=2 time vector A points to the right and has magnitude B = 2 A = 3.0. Vector C = -2 times vector A and has magnitude B = 2.0. Figure b shows vector A points to the right and has magnitude A=1.5. Vector B is shown below vector A, with their tails aligned. Vector B points to the right and has magnitude 2.0. In another view, Vector A is shown with vector B starting at the head of A and extending further to the right. Below them is a vector, labeled as vector R = vector A plus vector B, pointing to the right whose tail is aligned with the tail of vector A and whose head is aligned with the head of vector B. The magnitude of vector R is equal to magnitude A plus magnitude B = 3.5. Figure c shows vector A points to the right and has magnitude A=1.5. Vector B is shown below vector A, with their tails aligned. Vector minus B points to the right and has magnitude 3.2. In another view, Vector A is shown with vector minus B pointing to the left and with its head meeting the head of vector A. Below them is a vector, labeled as vector D = vector A minus vector B, shorter than B and pointing to the left whose head is aligned with the head of vector B. The magnitude of vector D is equal to magnitude of quantity A minus B = 1.7.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167133579584\">Now suppose your fishing buddy departs from point <em>A<\/em> (the campsite), walking in the direction to point <em>B<\/em> (the fishing hole), but he realizes he lost his tackle box when he stopped to rest at point <em>C<\/em> (located three-quarters of the distance between <em>A<\/em> and <em>B<\/em>, beginning from point <em>A<\/em>). So, he turns back and retraces his steps in the direction toward the campsite and finds the box lying on the path at some point <em>D<\/em> only 1.2 km away from point <em>C<\/em> (see <a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>(b)). What is his displacement vector [latex]{\\stackrel{\\to }{D}}_{AD}[\/latex] when he finds the box at point <em>D<\/em>? What is his displacement vector [latex]{\\stackrel{\\to }{D}}_{DB}[\/latex] from point <em>D<\/em> to the hole? We have already established that at rest point <em>C<\/em> his displacement vector is [latex]{\\stackrel{\\to }{D}}_{AC}=0.75{\\stackrel{\\to }{D}}_{AB}[\/latex]. Starting at point <em>C<\/em>, he walks southwest (toward the campsite), which means his new displacement vector [latex]{\\stackrel{\\to }{D}}_{CD}[\/latex] from point <em>C<\/em> to point <em>D<\/em> is antiparallel to [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex]. Its magnitude [latex]|{\\stackrel{\\to }{D}}_{CD}|[\/latex] is [latex]{D}_{CD}=1.2\\phantom{\\rule{0.2em}{0ex}}\\text{km}=0.2{D}_{AB}[\/latex], so his second displacement vector is [latex]{\\stackrel{\\to }{D}}_{CD}=-0.2{\\stackrel{\\to }{D}}_{AB}[\/latex]. His total displacement [latex]{\\stackrel{\\to }{D}}_{AD}[\/latex] relative to the campsite is the <span>vector sum<\/span> of the two displacement vectors: vector [latex]{\\stackrel{\\to }{D}}_{AC}[\/latex] (from the campsite to the rest point) and vector [latex]{\\stackrel{\\to }{D}}_{CD}[\/latex] (from the rest point to the point where he finds his box):<\/p>\n<div id=\"fs-id1167132225666\" class=\"equation-callout\">\n<div id=\"fs-id1167132476572\">[latex]{\\stackrel{\\to }{D}}_{AD}={\\stackrel{\\to }{D}}_{AC}+{\\stackrel{\\to }{D}}_{CD}.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167132507760\">The vector sum of two (or more) vectors is called the <span>resultant vector<\/span> or, for short, the <em>resultant<\/em>. When the vectors on the right-hand-side of <a href=\"#fs-id1167132476572\" class=\"autogenerated-content\">(Figure)<\/a> are known, we can find the resultant [latex]{\\stackrel{\\to }{D}}_{AD}[\/latex] as follows:<\/p>\n<div id=\"fs-id1167133527385\">[latex]{\\stackrel{\\to }{D}}_{AD}={\\stackrel{\\to }{D}}_{AC}+{\\stackrel{\\to }{D}}_{CD}=0.75{\\stackrel{\\to }{D}}_{AB}-0.2\\phantom{\\rule{0.2em}{0ex}}{\\stackrel{\\to }{D}}_{AB}=\\left(0.75-0.2\\right){\\stackrel{\\to }{D}}_{AB}=0.55{\\stackrel{\\to }{D}}_{AB}.[\/latex]<\/div>\n<p id=\"fs-id1167132416496\">When your friend finally reaches the pond at <em>B<\/em>, his displacement vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex] from point <em>A<\/em> is the vector sum of his displacement vector [latex]{\\stackrel{\\to }{D}}_{AD}[\/latex] from point <em>A<\/em> to point <em>D<\/em> and his displacement vector [latex]{\\stackrel{\\to }{D}}_{DB}[\/latex] from point <em>D<\/em> to the fishing hole: [latex]{\\stackrel{\\to }{D}}_{AB}={\\stackrel{\\to }{D}}_{AD}+{\\stackrel{\\to }{D}}_{DB}[\/latex] (see <a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>(c)). This means his displacement vector [latex]{\\stackrel{\\to }{D}}_{DB}[\/latex] is the <span>difference of two vectors<\/span>:<\/p>\n<div id=\"fs-id1167132441548\">[latex]{\\stackrel{\\to }{D}}_{DB}={\\stackrel{\\to }{D}}_{AB}-{\\stackrel{\\to }{D}}_{AD}={\\stackrel{\\to }{D}}_{AB}+\\left(\\text{\u2212}{\\stackrel{\\to }{D}}_{AD}\\right).[\/latex]<\/div>\n<p id=\"fs-id1167132281683\">Notice that a difference of two vectors is nothing more than a vector sum of two vectors because the second term in <a href=\"#fs-id1167132441548\" class=\"autogenerated-content\">(Figure)<\/a> is vector [latex]\\text{\u2212}{\\stackrel{\\to }{D}}_{AD}[\/latex] (which is antiparallel to [latex]{\\stackrel{\\to }{D}}_{AD}\\right)[\/latex]. When we substitute <a href=\"#fs-id1167133527385\" class=\"autogenerated-content\">(Figure)<\/a> into <a href=\"#fs-id1167132441548\" class=\"autogenerated-content\">(Figure)<\/a>, we obtain the second displacement vector:<\/p>\n<div id=\"fs-id1167133768313\">[latex]{\\stackrel{\\to }{D}}_{DB}={\\stackrel{\\to }{D}}_{AB}-{\\stackrel{\\to }{D}}_{AD}={\\stackrel{\\to }{D}}_{AB}-0.55{\\stackrel{\\to }{D}}_{AB}=\\left(1.0-0.55\\right){\\stackrel{\\to }{D}}_{AB}=0.45{\\stackrel{\\to }{D}}_{AB}.[\/latex]<\/div>\n<p id=\"fs-id1167132499982\">This result means your friend walked [latex]{D}_{DB}=0.45{D}_{AB}=0.45\\left(6.0\\phantom{\\rule{0.2em}{0ex}}\\text{km}\\right)=2.7\\phantom{\\rule{0.2em}{0ex}}\\text{km}[\/latex] from the point where he finds his tackle box to the fishing hole.<\/p>\n<p id=\"fs-id1167132237953\">When vectors [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}[\/latex] lie along a line (that is, in one dimension), such as in the camping example, their resultant [latex]\\stackrel{\\to }{R}=\\stackrel{\\to }{A}+\\stackrel{\\to }{B}[\/latex] and their difference [latex]\\stackrel{\\to }{D}=\\stackrel{\\to }{A}-\\stackrel{\\to }{B}[\/latex] both lie along the same direction. We can illustrate the addition or subtraction of vectors by drawing the corresponding vectors to scale in one dimension, as shown in <a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1167132378007\">To illustrate the resultant when [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}[\/latex] are two parallel vectors, we draw them along one line by placing the origin of one vector at the end of the other vector in head-to-tail fashion (see <a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>(b)). The magnitude of this resultant is the sum of their magnitudes: <em>R<\/em> = <em>A<\/em> + <em>B<\/em>. The direction of the resultant is parallel to both vectors. When vector [latex]\\stackrel{\\to }{A}[\/latex] is antiparallel to vector [latex]\\stackrel{\\to }{B}[\/latex], we draw them along one line in either head-to-head fashion (<a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>(c)) or tail-to-tail fashion. The magnitude of the vector difference, then, is the <em>absolute value<\/em> [latex]D=|A-B|[\/latex] of the difference of their magnitudes. The direction of the difference vector [latex]\\stackrel{\\to }{D}[\/latex] is parallel to the direction of the longer vector.<\/p>\n<p id=\"fs-id1167132299184\">In general, in one dimension\u2014as well as in higher dimensions, such as in a plane or in space\u2014we can add any number of vectors and we can do so in any order because the addition of vectors is <span>commutative<\/span>,<\/p>\n<div id=\"fs-id1167132463322\" class=\"equation-callout\">\n<div id=\"fs-id1167132501766\">[latex]\\stackrel{\\to }{A}+\\stackrel{\\to }{B}=\\stackrel{\\to }{B}+\\stackrel{\\to }{A},[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167132569455\">and <span>associative<\/span>,<\/p>\n<div id=\"fs-id1167132239544\" class=\"equation-callout\">\n<div id=\"fs-id1167133454633\">[latex]\\left(\\stackrel{\\to }{A}+\\stackrel{\\to }{B}\\right)+\\stackrel{\\to }{C}=\\stackrel{\\to }{A}+\\left(\\stackrel{\\to }{B}+\\stackrel{\\to }{C}\\right).[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167132342483\">Moreover, multiplication by a scalar is <span>distributive<\/span>:<\/p>\n<div id=\"fs-id1167132615617\" class=\"equation-callout\">\n<div id=\"fs-id1167132615620\">[latex]{\\alpha }_{1}\\stackrel{\\to }{A}+{\\alpha }_{2}\\stackrel{\\to }{A}=\\left({\\alpha }_{1}+{\\alpha }_{2}\\right)\\stackrel{\\to }{A}.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167132413323\">We used the distributive property in <a href=\"#fs-id1167133527385\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1167133768313\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1167132397556\">When adding many vectors in one dimension, it is convenient to use the concept of a <span>unit vector<\/span>. A unit vector, which is denoted by a letter symbol with a hat, such as [latex]\\stackrel{^}{u}[\/latex], has a magnitude of one and does not have any physical unit so that [latex]|\\stackrel{^}{u}|\\equiv u=1[\/latex]. The only role of a unit vector is to specify direction. For example, instead of saying vector [latex]{\\stackrel{\\to }{D}}_{AB}[\/latex] has a magnitude of 6.0 km and a direction of northeast, we can introduce a unit vector [latex]\\stackrel{^}{u}[\/latex] that points to the northeast and say succinctly that [latex]{\\stackrel{\\to }{D}}_{AB}=\\left(6.0\\phantom{\\rule{0.2em}{0ex}}\\text{km}\\right)\\stackrel{^}{u}[\/latex]. Then the southwesterly direction is simply given by the unit vector [latex]\\text{\u2212}\\stackrel{^}{u}[\/latex]. In this way, the displacement of 6.0 km in the southwesterly direction is expressed by the vector<\/p>\n<div id=\"fs-id1167132499315\" class=\"unnumbered\">[latex]{\\stackrel{\\to }{D}}_{BA}=\\left(-6.0\\phantom{\\rule{0.2em}{0ex}}\\text{km}\\right)\\stackrel{^}{u}.[\/latex]<\/div>\n<div id=\"fs-id1167132203045\" class=\"textbox examples\">\n<p id=\"fs-id1167132203047\"><span>A Ladybug Walker<\/span><br>\nA long measuring stick rests against a wall in a physics laboratory with its 200-cm end at the floor. A ladybug lands on the 100-cm mark and crawls randomly along the stick. It first walks 15 cm toward the floor, then it walks 56 cm toward the wall, then it walks 3 cm toward the floor again. Then, after a brief stop, it continues for 25 cm toward the floor and then, again, it crawls up 19 cm toward the wall before coming to a complete rest (<a href=\"#CNX_UPhysics_02_01_ladybug\" class=\"autogenerated-content\">(Figure)<\/a>). Find the vector of its total displacement and its final resting position on the stick.<\/p>\n<p id=\"fs-id1167132688912\"><span>Strategy<\/span><br>\nIf we choose the direction along the stick toward the floor as the direction of unit vector [latex]\\stackrel{^}{u}[\/latex], then the direction toward the floor is [latex]+\\stackrel{^}{u}[\/latex] and the direction toward the wall is [latex]\\text{\u2212}\\stackrel{^}{u}[\/latex]. The ladybug makes a total of five displacements:<\/p>\n<div id=\"fs-id1167133508498\" class=\"unnumbered\">[latex]\\begin{array}{c}{\\stackrel{\\to }{D}}_{1}=\\left(15\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(+\\stackrel{^}{u}\\right),\\hfill \\\\ {\\stackrel{\\to }{D}}_{2}=\\left(56\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(\\text{\u2212}\\stackrel{^}{u}\\right),\\hfill \\\\ {\\stackrel{\\to }{D}}_{3}=\\left(3\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(+\\stackrel{^}{u}\\right),\\hfill \\\\ {\\stackrel{\\to }{D}}_{4}=\\left(25\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(+\\stackrel{^}{u}\\right),\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\hfill \\\\ {\\stackrel{\\to }{D}}_{5}=\\left(19\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(\\text{\u2212}\\stackrel{^}{u}\\right).\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167132241640\">The total displacement [latex]\\stackrel{\\to }{D}[\/latex] is the resultant of all its displacement vectors.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_ladybug\">\n<div class=\"bc-figcaption figcaption\">Five displacements of the ladybug. Note that in this schematic drawing, magnitudes of displacements are not drawn to scale. (credit: modification of work by \u201cPersian Poet Gal\u201d\/Wikimedia Commons)<\/div>\n<p><span id=\"fs-id1167133748196\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_ladybug.jpg\" alt=\"Five illustrations of a ladybug on a ruler leaning against a wall. The +u hat direction is toward the floor parallel to the ruler, and the \u2013 u hat direction is up along the ruler. In the first illustration, the ladybug is located near the middle of the ruler and vector D sub 1 points down the ruler. In the second illustration, the ladybug is located lower, where the head of vector D sub 1 is in the first illustration, and vector D sub 2 points up the ruler. In the third illustration, the ladybug is located higher, where the head of vector D sub 2 is in the second illustration, and vector D sub 3 points down the ruler. In the fourth illustration, the ladybug is located lower, where the head of vector D sub 3 is in the third illustration, and vector D sub 4 points down the ruler. In the fifth illustration, the ladybug is located lower, where the head of vector D sub 4 is in the fourth illustration, and vector D sub 5 points up the ruler.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167133826776\"><span>Solution<\/span><br>\nThe resultant of all the displacement vectors is<\/p>\n<div id=\"fs-id1167133826781\" class=\"unnumbered\">[latex]\\begin{array}{cc}\\hfill \\stackrel{\\to }{D}&amp; ={\\stackrel{\\to }{D}}_{1}+{\\stackrel{\\to }{D}}_{2}+{\\stackrel{\\to }{D}}_{3}+{\\stackrel{\\to }{D}}_{4}+{\\stackrel{\\to }{D}}_{5}\\hfill \\\\ &amp; =\\left(15\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(+\\stackrel{^}{u}\\right)+\\left(56\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(\\text{\u2212}\\stackrel{^}{u}\\right)+\\left(3\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(+\\stackrel{^}{u}\\right)+\\left(25\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(+\\stackrel{^}{u}\\right)+\\left(19\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)\\left(\\text{\u2212}\\stackrel{^}{u}\\right)\\hfill \\\\ &amp; =\\left(15-56+3+25-19\\right)\\text{cm}\\stackrel{^}{u}\\hfill \\\\ &amp; =-32\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\stackrel{^}{u}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167132436133\">In this calculation, we use the distributive law given by <a href=\"#fs-id1167132615620\" class=\"autogenerated-content\">(Figure)<\/a>. The result reads that the total displacement vector points away from the 100-cm mark (initial landing site) toward the end of the meter stick that touches the wall. The end that touches the wall is marked 0 cm, so the final position of the ladybug is at the (100 \u2013 32)cm = 68-cm mark.<\/p>\n<\/div>\n<div id=\"fs-id1167133521609\" class=\"check-understanding\">\n<div id=\"fs-id1167132473641\">\n<div id=\"fs-id1167132473643\">\n<p id=\"fs-id1167132473645\"><strong>Check Your Understanding<\/strong> A cave diver enters a long underwater tunnel. When her displacement with respect to the entry point is 20 m, she accidentally drops her camera, but she doesn\u2019t notice it missing until she is some 6 m farther into the tunnel. She swims back 10 m but cannot find the camera, so she decides to end the dive. How far from the entry point is she? Taking the positive direction out of the tunnel, what is her displacement vector relative to the entry point?<\/p>\n<\/div>\n<div id=\"fs-id1167133773324\">\n<p id=\"fs-id1167133773326\">16 m; [latex]\\stackrel{\\to }{D}=-16\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\stackrel{^}{u}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1167132371115\">\n<h3>Algebra of Vectors in Two Dimensions<\/h3>\n<p id=\"fs-id1167132342179\">When vectors lie in a plane\u2014that is, when they are in two dimensions\u2014they can be multiplied by scalars, added to other vectors, or subtracted from other vectors in accordance with the general laws expressed by <a href=\"#fs-id1167133686173\" class=\"autogenerated-content\">(Figure)<\/a>, <a href=\"#fs-id1167133394048\" class=\"autogenerated-content\">(Figure)<\/a>, <a href=\"#fs-id1167132501766\" class=\"autogenerated-content\">(Figure)<\/a>, and <a href=\"#fs-id1167133454633\" class=\"autogenerated-content\">(Figure)<\/a>. However, the addition rule for two vectors in a plane becomes more complicated than the rule for vector addition in one dimension. We have to use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. This geometric approach is commonly used in navigation (<a href=\"#CNX_UPhysics_02_01_navigation\" class=\"autogenerated-content\">(Figure)<\/a>). In this section, we need to have at hand two rulers, a triangle, a protractor, a pencil, and an eraser for drawing vectors to scale by geometric constructions.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_navigation\">\n<div class=\"bc-figcaption figcaption\">In navigation, the laws of geometry are used to draw resultant displacements on nautical maps.<\/div>\n<p><span id=\"fs-id1167133541239\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_navigation.jpg\" alt=\"A photograph of someone measuring distance on a map using calipers and a ruler.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167132594682\">For a geometric construction of the sum of two vectors in a plane, we follow <span>the parallelogram rule<\/span>. Suppose two vectors [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}[\/latex] are at the arbitrary positions shown in <a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>. Translate either one of them in parallel to the beginning of the other vector, so that after the translation, both vectors have their origins at the same point. Now, at the end of vector [latex]\\stackrel{\\to }{A}[\/latex] we draw a line parallel to vector [latex]\\stackrel{\\to }{B}[\/latex] and at the end of vector [latex]\\stackrel{\\to }{B}[\/latex] we draw a line parallel to vector [latex]\\stackrel{\\to }{A}[\/latex] (the dashed lines in <a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>). In this way, we obtain a parallelogram. From the origin of the two vectors we draw a diagonal that is the resultant [latex]\\stackrel{\\to }{R}[\/latex] of the two vectors: [latex]\\stackrel{\\to }{R}=\\stackrel{\\to }{A}+\\stackrel{\\to }{B}[\/latex] (<a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>(a)). The other diagonal of this parallelogram is the vector difference of the two vectors [latex]\\stackrel{\\to }{D}=\\stackrel{\\to }{A}-\\stackrel{\\to }{B}[\/latex], as shown in <a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>(b). Notice that the end of the difference vector is placed at the end of vector [latex]\\stackrel{\\to }{A}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector04\">\n<div class=\"bc-figcaption figcaption\">The parallelogram rule for the addition of two vectors. Make the parallel translation of each vector to a point where their origins (marked by the dot) coincide and construct a parallelogram with two sides on the vectors and the other two sides (indicated by dashed lines) parallel to the vectors. (a) Draw the resultant vector [latex]\\stackrel{\\to }{R}[\/latex] along the diagonal of the parallelogram from the common point to the opposite corner. Length <em>R<\/em> of the resultant vector is <em>not<\/em> equal to the sum of the magnitudes of the two vectors. (b) Draw the difference vector [latex]\\stackrel{\\to }{D}=\\stackrel{\\to }{A}-\\stackrel{\\to }{B}[\/latex] along the diagonal connecting the ends of the vectors. Place the origin of vector [latex]\\stackrel{\\to }{D}[\/latex] at the end of vector [latex]\\stackrel{\\to }{B}[\/latex] and the end (arrowhead) of vector [latex]\\stackrel{\\to }{D}[\/latex] at the end of vector [latex]\\stackrel{\\to }{A}[\/latex]. Length <em>D<\/em> of the difference vector is <em>not<\/em> equal to the difference of magnitudes of the two vectors.<\/div>\n<p><span id=\"fs-id1167133635872\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector04.jpg\" alt=\"The parallelogram method for adding vectors is illustrated. In figure a, vectors A and B are shown. Vector A points to the right and down and vector B points right and up. Vectors A and B are then shown as solid arrows with their tails together, and their directions as before. A dashed line parallel to vector A but shifted so it starts at the head of B is shown. A second dashed line, parallel to B and starting at the head of A is also shown. The vectors A and B and the two dashed lines form a parallelogram. A third vector, labeled vector R = vector A plus vector B, is shown. The tail of vector R is at the tails of vectors A and B, and the head of vector R is where the dashed lines meet each other, diagonally across the parallelogram. We note that the magnitude of R is not equal to the magnitude of A plus the magnitude of B. In figure b, vectors A and minus B are shown. Vector minus B is vector B from part a, rotated 180 degrees. Vector A points to the right and down and vector minus B points left and down. Vectors A and B are then shown as solid arrows with their tails together, and their directions as before. A dashed line parallel to vector A but shifted so it starts at the head of B is shown. A second dashed line, parallel to B and starting at the head of A is also shown. The vectors A and B and the two dashed lines form a parallelogram. A third vector, labeled vector D is shown. The tail of vector D is at the head of vector B, and the head of vector D is at the head of vector A, diagonally across the parallelogram. We note that vector D is equal to vector A minus vector B, but the magnitude of D is not equal to the magnitude of A minus the B.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167132362973\">It follows from the parallelogram rule that neither the magnitude of the resultant vector nor the magnitude of the difference vector can be expressed as a simple sum or difference of magnitudes <em>A<\/em> and <em>B<\/em>, because the length of a diagonal cannot be expressed as a simple sum of side lengths. When using a geometric construction to find magnitudes [latex]|\\stackrel{\\to }{R}|[\/latex] and [latex]|\\stackrel{\\to }{D}|[\/latex], we have to use trigonometry laws for triangles, which may lead to complicated algebra. There are two ways to circumvent this algebraic complexity. One way is to use the method of components, which we examine in the next section. The other way is to draw the vectors to scale, as is done in navigation, and read approximate vector lengths and angles (directions) from the graphs. In this section we examine the second approach.<\/p>\n<p id=\"fs-id1167132501642\">If we need to add three or more vectors, we repeat the parallelogram rule for the pairs of vectors until we find the resultant of all of the resultants. For three vectors, for example, we first find the resultant of vector 1 and vector 2, and then we find the resultant of this resultant and vector 3. The order in which we select the pairs of vectors does not matter because the operation of vector addition is commutative and associative (see <a href=\"#fs-id1167132501766\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1167133454633\" class=\"autogenerated-content\">(Figure)<\/a>). Before we state a general rule that follows from repetitive applications of the parallelogram rule, let\u2019s look at the following example.<\/p>\n<p id=\"fs-id1167132267118\">Suppose you plan a vacation trip in Florida. Departing from Tallahassee, the state capital, you plan to visit your uncle Joe in Jacksonville, see your cousin Vinny in Daytona Beach, stop for a little fun in Orlando, see a circus performance in Tampa, and visit the University of Florida in Gainesville. Your route may be represented by five displacement vectors [latex]\\stackrel{\\to }{A},[\/latex] [latex]\\stackrel{\\to }{B}[\/latex], [latex]\\stackrel{\\to }{C}[\/latex], [latex]\\stackrel{\\to }{D}[\/latex], and [latex]\\stackrel{\\to }{E}[\/latex], which are indicated by the red vectors in <a href=\"#CNX_UPhysics_02_01_vector05\" class=\"autogenerated-content\">(Figure)<\/a>. What is your total displacement when you reach Gainesville? The total displacement is the vector sum of all five displacement vectors, which may be found by using the parallelogram rule four times. Alternatively, recall that the displacement vector has its beginning at the initial position (Tallahassee) and its end at the final position (Gainesville), so the total displacement vector can be drawn directly as an arrow connecting Tallahassee with Gainesville (see the green vector in <a href=\"#CNX_UPhysics_02_01_vector05\" class=\"autogenerated-content\">(Figure)<\/a>). When we use the parallelogram rule four times, the resultant [latex]\\stackrel{\\to }{R}[\/latex] we obtain is exactly this green vector connecting Tallahassee with Gainesville: [latex]\\stackrel{\\to }{R}=\\stackrel{\\to }{A}+\\stackrel{\\to }{B}+\\stackrel{\\to }{C}+\\stackrel{\\to }{D}+\\stackrel{\\to }{E}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector05\">\n<div class=\"bc-figcaption figcaption\">When we use the parallelogram rule four times, we obtain the resultant vector [latex]\\stackrel{\\to }{R}=\\stackrel{\\to }{A}+\\stackrel{\\to }{B}+\\stackrel{\\to }{C}+\\stackrel{\\to }{D}+\\stackrel{\\to }{E}[\/latex], which is the green vector connecting Tallahassee with Gainesville.<\/div>\n<p><span id=\"fs-id1167132322121\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector05.jpg\" alt=\"A map of Florida with the following vectors shown in red: Vector A from Tallahassee to Jacksonville, almost due west. Vector B from Jacksonville to Daytona Beach, southeast. Vector C from Daytona Beach to Orlando, southwest. Vector D from Orlando to Tampa, southwest (but less vertical than vector C). Vector E from Tampa to Gainesville, slightly east of north. Vector R from Tallahassee to Gainsville is shown as a green arrow.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167132444404\">Drawing the resultant vector of many vectors can be generalized by using the following <span>tail-to-head geometric construction<\/span>. Suppose we want to draw the resultant vector [latex]\\stackrel{\\to }{R}[\/latex] of four vectors [latex]\\stackrel{\\to }{A}[\/latex], [latex]\\stackrel{\\to }{B}[\/latex], [latex]\\stackrel{\\to }{C}[\/latex], and [latex]\\stackrel{\\to }{D}[\/latex] (<a href=\"#CNX_UPhysics_02_01_vector06\" class=\"autogenerated-content\">(Figure)<\/a>(a)). We select any one of the vectors as the first vector and make a parallel translation of a second vector to a position where the origin (\u201ctail\u201d) of the second vector coincides with the end (\u201chead\u201d) of the first vector. Then, we select a third vector and make a parallel translation of the third vector to a position where the origin of the third vector coincides with the end of the second vector. We repeat this procedure until all the vectors are in a head-to-tail arrangement like the one shown in <a href=\"#CNX_UPhysics_02_01_vector06\" class=\"autogenerated-content\">(Figure)<\/a>. We draw the resultant vector [latex]\\stackrel{\\to }{R}[\/latex] by connecting the origin (\u201ctail\u201d) of the first vector with the end (\u201chead\u201d) of the last vector. The end of the resultant vector is at the end of the last vector. Because the addition of vectors is associative and commutative, we obtain the same resultant vector regardless of which vector we choose to be first, second, third, or fourth in this construction.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector06\">\n<div class=\"bc-figcaption figcaption\">Tail-to-head method for drawing the resultant vector [latex]\\stackrel{\\to }{R}=\\stackrel{\\to }{A}+\\stackrel{\\to }{B}+\\stackrel{\\to }{C}+\\stackrel{\\to }{D}[\/latex]. (a) Four vectors of different magnitudes and directions. (b) Vectors in (a) are translated to new positions where the origin (\u201ctail\u201d) of one vector is at the end (\u201chead\u201d) of another vector. The resultant vector is drawn from the origin (\u201ctail\u201d) of the first vector to the end (\u201chead\u201d) of the last vector in this arrangement.<\/div>\n<p><span id=\"fs-id1167132205226\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector06.jpg\" alt=\"In figure a, four vectors, labeled A, B, C, and D are shown individually. In figure b, the vectors are shown arranged head to tail: Vector A\u2019s tail is at the head of D. Vector C\u2019s tail is at the head of A. And vector B\u2019s tail is at the head of C. Each vector is pointing in the same direction as it is in figure a. A fifth vector, R, starts at the tail of vector D and ends at the head of vector B.\"><\/span><\/p><\/div>\n<div id=\"fs-id1167132310249\" class=\"textbox examples\">\n<p id=\"fs-id1167132546290\"><span>Geometric Construction of the Resultant<\/span><br>\nThe three displacement vectors [latex]\\stackrel{\\to }{A}[\/latex], [latex]\\stackrel{\\to }{B}[\/latex], and [latex]\\stackrel{\\to }{C}[\/latex] in <a href=\"#CNX_UPhysics_02_01_vector07\" class=\"autogenerated-content\">(Figure)<\/a> are specified by their magnitudes <em>A<\/em> = 10.0, <em>B<\/em> = 7.0, and <em>C<\/em> = 8.0, respectively, and by their respective direction angles with the horizontal direction [latex]\\alpha =35\\text{\u00b0}[\/latex], [latex]\\beta =-110\\text{\u00b0}[\/latex], and [latex]\\gamma =30\\text{\u00b0}[\/latex]. The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) [latex]\\stackrel{\\to }{R}=\\stackrel{\\to }{A}+\\stackrel{\\to }{B}[\/latex], (b) [latex]\\stackrel{\\to }{D}=\\stackrel{\\to }{A}-\\stackrel{\\to }{B}\\text{, and}[\/latex] (c) [latex]\\stackrel{\\to }{S}=\\stackrel{\\to }{A}-3\\stackrel{\\to }{B}+\\stackrel{\\to }{C}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector07\">\n<div class=\"bc-figcaption figcaption\">Vectors used in <a href=\"#fs-id1167132310249\" class=\"autogenerated-content\">(Figure)<\/a> and in the Check Your Understanding feature that follows.<\/div>\n<p><span id=\"fs-id1167132361887\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector07.jpg\" alt=\"Vector A has magnitude 10.0 and is at an angle alpha = 35 degrees counterclockwise from the horizontal. It points up and right. Vector B has magnitude 7.0 and is at an angle beta = -110 degrees clockwise from the horizontal. It points down and left. Vector C has magnitude 8.0 and is at an angle gamma = 30 degrees counterclockwise from the horizontal. It points up and right. Vector F has magnitude 20.0 and is at an angle phi = 110 degrees counterclockwise from the horizontal. It points up and left.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167132688124\"><span>Strategy<\/span><br>\nIn geometric construction, to find a vector means to find its magnitude and its direction angle with the horizontal direction. The strategy is to draw to scale the vectors that appear on the right-hand side of the equation and construct the resultant vector. Then, use a ruler and a protractor to read the magnitude of the resultant and the direction angle. For parts (a) and (b) we use the parallelogram rule. For (c) we use the tail-to-head method.<\/p>\n<p id=\"fs-id1167132688134\"><span>Solution<\/span><br>\nFor parts (a) and (b), we attach the origin of vector [latex]\\stackrel{\\to }{B}[\/latex] to the origin of vector [latex]\\stackrel{\\to }{A}[\/latex], as shown in <a href=\"#CNX_UPhysics_02_01_vector08\" class=\"autogenerated-content\">(Figure)<\/a>, and construct a parallelogram. The shorter diagonal of this parallelogram is the sum [latex]\\stackrel{\\to }{A}+\\stackrel{\\to }{B}[\/latex]. The longer of the diagonals is the difference [latex]\\stackrel{\\to }{A}-\\stackrel{\\to }{B}[\/latex]. We use a ruler to measure the lengths of the diagonals, and a protractor to measure the angles with the horizontal. For the resultant [latex]\\stackrel{\\to }{R}[\/latex], we obtain <em>R<\/em> = 5.8 cm and [latex]{\\theta }_{R}\\approx 0\\text{\u00b0}[\/latex]. For the difference [latex]\\stackrel{\\to }{D}[\/latex], we obtain <em>D<\/em> = 16.2 cm and [latex]{\\theta }_{D}=49.3\\text{\u00b0}[\/latex], which are shown in <a href=\"#CNX_UPhysics_02_01_vector08\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector08\">\n<div class=\"bc-figcaption figcaption\">Using the parallelogram rule to solve (a) (finding the resultant, red) and (b) (finding the difference, blue).<\/div>\n<p><span id=\"fs-id1167132685555\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector08.jpg\" alt=\"Three diagrams of vectors A and B. Vectors A and B are shown placed tail to tail. Vector A points up and right and has magnitude 10.0. Vector B points down and left and has magnitude 7.0. The angle between vectors A and B is 145 degrees. In the second diagram, Vectors A and B are shown again along with the dashed lines completing the parallelogram. Vector R equaling the sum of vectors A and B is shown as the vector from the tails of A and B to the opposite vertex of the parallelogram. The magnitude of R is 5.8. In the third diagram, Vectors A and B are shown again along with the dashed lines completing the parallelogram. Vector D equaling the difference of vectors A and B is shown as the vector from the head of B to the head of A. The magnitude of D is 16.2, and the angle between D and the horizontal is 49.3 degrees. Vector R in the second diagram is much shorter than vector D in the third diagram.\"><\/span><\/p><\/div>\n<p id=\"fs-id1167132451912\">For (c), we can start with vector [latex]-3\\stackrel{\\to }{B}[\/latex] and draw the remaining vectors tail-to-head as shown in <a href=\"#CNX_UPhysics_02_01_vector09\" class=\"autogenerated-content\">(Figure)<\/a>. In vector addition, the order in which we draw the vectors is unimportant, but drawing the vectors to scale is very important. Next, we draw vector [latex]\\stackrel{\\to }{S}[\/latex] from the origin of the first vector to the end of the last vector and place the arrowhead at the end of [latex]\\stackrel{\\to }{S}[\/latex]. We use a ruler to measure the length of [latex]\\stackrel{\\to }{S}[\/latex], and find that its magnitude is<\/p>\n<div><\/div>\n<p><em>S<\/em> = 36.9 cm. We use a protractor and find that its direction angle is [latex]{\\theta }_{S}=52.9\\text{\u00b0}[\/latex]. This solution is shown in <a href=\"#CNX_UPhysics_02_01_vector09\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector09\">\n<div class=\"bc-figcaption figcaption\">Using the tail-to-head method to solve (c) (finding vector [latex]\\stackrel{\\to }{S}[\/latex], green).<\/div>\n<p><span id=\"fs-id1167132499445\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector09.jpg\" alt=\"Three vectors are shown in blue and placed head to tail: Vector minus 3 B points up and right and has magnitude 3 B = 21.0. Vector A starts at the head of B, points up and right, and has a magnitude of A=10.0. The angle between vector A and vector minus 3 B is 145 degrees. Vector C starts at the head of A and has magnitude C=8.0. Vector S is green and goes from the tail of minus 3 B to the head of C. Vector S equals vector A minus 3 vector B plus vector C, has a magnitude of S=36.9 and makes an angle of 52.9 degrees counterclockwise with the horizontal.\"><\/span><\/p><\/div>\n<\/div>\n<div id=\"fs-id1167132487190\" class=\"check-understanding\">\n<div id=\"fs-id1167132487194\">\n<div id=\"fs-id1167132487196\">\n<p id=\"fs-id1167133787376\"><strong>Check Your Understanding<\/strong> Using the three displacement vectors [latex]\\stackrel{\\to }{A}[\/latex], [latex]\\stackrel{\\to }{B}[\/latex], and [latex]\\stackrel{\\to }{F}[\/latex] in <a href=\"#CNX_UPhysics_02_01_vector07\" class=\"autogenerated-content\">(Figure)<\/a>, choose a convenient scale, and use a ruler and a protractor to find vector [latex]\\stackrel{\\to }{G}[\/latex] given by the vector equation [latex]\\stackrel{\\to }{G}=\\stackrel{\\to }{A}+2\\stackrel{\\to }{B}-\\stackrel{\\to }{F}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1167132326222\">\n<p id=\"fs-id1167132576679\"><em>G<\/em> = 28.2 cm, [latex]{\\theta }_{G}=291\\text{\u00b0}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133568457\" class=\"media-2\">\n<p id=\"fs-id1167133568461\">Observe the addition of vectors in a plane by visiting this <a href=\"https:\/\/openstaxcollege.org\/l\/21compveccalc\">vector calculator<\/a> and this <a href=\"https:\/\/openstaxcollege.org\/l\/21phetvecaddsim\">Phet simulation<\/a>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\" id=\"fs-id1167133772309\">\n<h3>Summary<\/h3>\n<ul id=\"fs-id1167132451699\">\n<li>A vector quantity is any quantity that has magnitude and direction, such as displacement or velocity. Vector quantities are represented by mathematical objects called vectors.<\/li>\n<li>Geometrically, vectors are represented by arrows, with the end marked by an arrowhead. The length of the vector is its magnitude, which is a positive scalar. On a plane, the direction of a vector is given by the angle the vector makes with a reference direction, often an angle with the horizontal. The direction angle of a vector is a scalar.<\/li>\n<li>Two vectors are equal if and only if they have the same magnitudes and directions. Parallel vectors have the same direction angles but may have different magnitudes. Antiparallel vectors have direction angles that differ by [latex]180\\text{\u00b0}[\/latex]. Orthogonal vectors have direction angles that differ by [latex]90\\text{\u00b0}[\/latex].<\/li>\n<li>When a vector is multiplied by a scalar, the result is another vector of a different length than the length of the original vector. Multiplication by a positive scalar does not change the original direction; only the magnitude is affected. Multiplication by a negative scalar reverses the original direction. The resulting vector is antiparallel to the original vector. Multiplication by a scalar is distributive. Vectors can be divided by nonzero scalars but cannot be divided by vectors.<\/li>\n<li>Two or more vectors can be added to form another vector. The vector sum is called the resultant vector. We can add vectors to vectors or scalars to scalars, but we cannot add scalars to vectors. Vector addition is commutative and associative.<\/li>\n<li>To construct a resultant vector of two vectors in a plane geometrically, we use the parallelogram rule. To construct a resultant vector of many vectors in a plane geometrically, we use the tail-to-head method.<\/li>\n<\/ul>\n<\/div>\n<div class=\"review-conceptual-questions\" id=\"fs-id1167132744637\">\n<h3>Conceptual Questions<\/h3>\n<div id=\"fs-id1167132744645\">\n<div id=\"fs-id1167132744647\">\n<p id=\"fs-id1167133708383\">A weather forecast states the temperature is predicted to be [latex]-5\\phantom{\\rule{0.2em}{0ex}}\\text{\u00b0}\\text{C}[\/latex] the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1167133766192\">\n<p id=\"fs-id1167133766194\">scalar<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133578427\">\n<div id=\"fs-id1167133578429\">\n<p id=\"fs-id1167133578431\">Which of the following is a vector: a person\u2019s height, the altitude on Mt. Everest, the velocity of a fly, the age of Earth, the boiling point of water, the cost of a book, Earth\u2019s population, or the acceleration of gravity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132458328\">\n<div id=\"fs-id1167132458330\">\n<p id=\"fs-id1167132458332\">Give a specific example of a vector, stating its magnitude, units, and direction.<\/p>\n<\/div>\n<div id=\"fs-id1167132483960\">\n<p id=\"fs-id1167132483962\">answers may vary<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132483967\">\n<div id=\"fs-id1167132483969\">\n<p id=\"fs-id1167132230862\">What do vectors and scalars have in common? How do they differ?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132352997\">\n<div id=\"fs-id1167132352999\">\n<p id=\"fs-id1167132353001\">Suppose you add two vectors [latex]\\stackrel{\\to }{A}[\/latex] and [latex]\\stackrel{\\to }{B}[\/latex]. What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?<\/p>\n<\/div>\n<div id=\"fs-id1167133640324\">\n<p id=\"fs-id1167133640326\">parallel, sum of magnitudes, antiparallel, zero<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133640332\">\n<div id=\"fs-id1167133640334\">\n<p id=\"fs-id1167133640336\">Is it possible to add a scalar quantity to a vector quantity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132315789\">\n<div id=\"fs-id1167132315791\">\n<p id=\"fs-id1167132415644\">Is it possible for two vectors of different magnitudes to add to zero? Is it possible for three vectors of different magnitudes to add to zero? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1167132415650\">\n<p id=\"fs-id1167132415652\">no, yes<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133635856\">\n<div id=\"fs-id1167133635858\">\n<p id=\"fs-id1167133635860\">Does the odometer in an automobile indicate a scalar or a vector quantity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167128860536\">\n<div id=\"fs-id1167128860538\">\n<p id=\"fs-id1167128860540\">When a 10,000-m runner competing on a 400-m track crosses the finish line, what is the runner\u2019s net displacement? Can this displacement be zero? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1167128860546\">\n<p id=\"fs-id1167132744080\">zero, yes<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132744085\">\n<div id=\"fs-id1167132744087\">\n<p id=\"fs-id1167132744089\">A vector has zero magnitude. Is it necessary to specify its direction? Explain.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132437699\">\n<div id=\"fs-id1167132437701\">\n<p id=\"fs-id1167132437703\">Can a magnitude of a vector be negative?<\/p>\n<\/div>\n<div id=\"fs-id1167132375593\">\n<p id=\"fs-id1167132375595\">no<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132375600\">\n<div id=\"fs-id1167132375603\">\n<p id=\"fs-id1167132341572\">Can the magnitude of a particle\u2019s displacement be greater that the distance traveled?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132318730\">\n<div id=\"fs-id1167132318732\">\n<p id=\"fs-id1167132318734\">If two vectors are equal, what can you say about their components? What can you say about their magnitudes? What can you say about their directions?<\/p>\n<\/div>\n<div id=\"fs-id1167132318740\">\n<p id=\"fs-id1167132318742\">equal, equal, the same<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132266747\">\n<div id=\"fs-id1167132266749\">\n<p id=\"fs-id1167132266751\">If three vectors sum up to zero, what geometric condition do they satisfy?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"review-problems\" id=\"fs-id1167132689349\">\n<h3>Problems<\/h3>\n<div id=\"fs-id1167132689356\">\n<div id=\"fs-id1167132255334\">\n<p id=\"fs-id1167132255336\">A scuba diver makes a slow descent into the depths of the ocean. His vertical position with respect to a boat on the surface changes several times. He makes the first stop 9.0 m from the boat but has a problem with equalizing the pressure, so he ascends 3.0 m and then continues descending for another 12.0 m to the second stop. From there, he ascends 4 m and then descends for 18.0 m, ascends again for 7 m and descends again for 24.0 m, where he makes a stop, waiting for his buddy. Assuming the positive direction up to the surface, express his net vertical displacement vector in terms of the unit vector. What is his distance to the boat?<\/p>\n<\/div>\n<div id=\"fs-id1167132255346\">\n<p id=\"fs-id1167133773306\">[latex]\\stackrel{\\to }{h}=-49\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\stackrel{^}{u}[\/latex], 49 m<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132471146\">\n<div id=\"fs-id1167132471148\">\n<p id=\"fs-id1167132471150\">In a tug-of-war game on one campus, 15 students pull on a rope at both ends in an effort to displace the central knot to one side or the other. Two students pull with force 196 N each to the right, four students pull with force 98 N each to the left, five students pull with force 62 N each to the left, three students pull with force 150 N each to the right, and one student pulls with force 250 N to the left. Assuming the positive direction to the right, express the net pull on the knot in terms of the unit vector. How big is the net pull on the knot? In what direction?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132559167\">\n<div id=\"fs-id1167132559169\">\n<p id=\"fs-id1167132559171\">Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point and what is the compass direction of a line connecting your starting point to your final position? Use a graphical method.<\/p>\n<\/div>\n<div id=\"fs-id1167132559177\">\n<p id=\"fs-id1167132559180\">30.8 m, [latex]35.7\\text{\u00b0}[\/latex] west of north<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132309641\">\n<div id=\"fs-id1167132296118\">\n<p id=\"fs-id1167132296120\">For the vectors given in the following figure, use a graphical method to find the following resultants: (a) [latex]\\stackrel{\\to }{A}+\\stackrel{\\to }{B}[\/latex], (b) [latex]\\stackrel{\\to }{C}+\\stackrel{\\to }{B}[\/latex], (c) [latex]\\stackrel{\\to }{D}+\\stackrel{\\to }{F}[\/latex], (d) [latex]\\stackrel{\\to }{A}-\\stackrel{\\to }{B}[\/latex], (e) [latex]\\stackrel{\\to }{D}-\\stackrel{\\to }{F}[\/latex], (f) [latex]\\stackrel{\\to }{A}+2\\stackrel{\\to }{F}[\/latex], (g); and (h) [latex]\\stackrel{\\to }{A}-4\\stackrel{\\to }{D}+2\\stackrel{\\to }{F}[\/latex].<\/p>\n<p><span id=\"fs-id1167132294810\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_problems_img.jpg\" alt=\"The x y coordinate system is shown, with positive x to the right and positive y up. Vector A has magnitude 10.0 and makes an angle of 30 degrees above the positive x direction. Vector B has magnitude 5.0 and makes an angle of 53 degrees above the positive x direction. Vector C has magnitude 12.0 and makes an angle of 60 degrees below the positive x direction. Vector D has magnitude 20.0 and makes an angle of 37 degrees above the negative x direction. Vector F has magnitude 20.0 and makes an angle of 30 degrees below the negative x direction.\"><\/span><\/p><\/div>\n<\/div>\n<div id=\"fs-id1167133740645\">\n<div id=\"fs-id1167133740647\">\n<p id=\"fs-id1167133740649\">A delivery man starts at the post office, drives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use a graphical method to find his net displacement vector.<\/p>\n<\/div>\n<div id=\"fs-id1167133740656\">\n<p id=\"fs-id1167133740658\">134 km, [latex]80\\text{\u00b0}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132741855\">\n<div id=\"fs-id1167132741858\">\n<p id=\"fs-id1167132741860\">An adventurous dog strays from home, runs three blocks east, two blocks north, one block east, one block north, and two blocks west. Assuming that each block is about 100 m, how far from home and in what direction is the dog? Use a graphical method.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132674455\">\n<div id=\"fs-id1167132674457\">\n<p id=\"fs-id1167132674459\">In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day and he is blown along the following directions: 2.50 km and [latex]45.0\\text{\u00b0}[\/latex] north of west, then 4.70 km and [latex]60.0\\text{\u00b0}[\/latex] south of east, then 1.30 km and [latex]25.0\\text{\u00b0}[\/latex] south of west, then 5.10 km straight east, then 1.70 km and [latex]5.00\\text{\u00b0}[\/latex] east of north, then 7.20 km and [latex]55.0\\text{\u00b0}[\/latex] south of west, and finally 2.80 km and [latex]10.0\\text{\u00b0}[\/latex] north of east. Use a graphical method to find the castaway\u2019s final position relative to the island.<\/p>\n<\/div>\n<div id=\"fs-id1167133722463\">\n<p id=\"fs-id1167133722465\">7.34 km, [latex]63.5\\text{\u00b0}[\/latex] south of east<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133766074\">\n<div id=\"fs-id1167133766076\">\n<p id=\"fs-id1167133766078\">A small plane flies 40.0 km in a direction [latex]60\\text{\u00b0}[\/latex] north of east and then flies 30.0 km in a direction [latex]15\\text{\u00b0}[\/latex] north of east. Use a graphical method to find the total distance the plane covers from the starting point and the direction of the path to the final position.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132546070\">\n<div id=\"fs-id1167132546072\">\n<p id=\"fs-id1167132546074\">A trapper walks a 5.0-km straight-line distance from his cabin to the lake, as shown in the following figure. Use a graphical method (the parallelogram rule) to determine the trapper\u2019s displacement directly to the east and displacement directly to the north that sum up to his resultant displacement vector. If the trapper walked only in directions east and north, zigzagging his way to the lake, how many kilometers would he have to walk to get to the lake?<\/p>\n<p><span id=\"fs-id1167132546077\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_problem3_img.jpg\" alt=\"North is up, east is to the right. A house and lake are shown. The x y coordiante system is also shown, with the origin near the house, the positive x direction to the right nad the positive y direction up. The vector from the house to the lake is shown as a straight red arrow, labeled as vector S, magnitude S=5.0 kilometers, and at an angle of 40 degrees above the posiitve x direction. Two meandering paths, path 1 and path 2, from the house to the lake are shown as dashed line.\"><\/span><\/p><\/div>\n<div id=\"fs-id1167132395714\">\n<p id=\"fs-id1167132395716\">3.8 km east, 3.2 km north, 7.0 km<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132395721\">\n<div id=\"fs-id1167132395723\">\n<p id=\"fs-id1167132395725\">A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is [latex]35\\text{\u00b0}[\/latex]. How wide is the river?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132526027\">\n<div id=\"fs-id1167132526029\">\n<p id=\"fs-id1167132526031\">A pedestrian walks 6.0 km east and then 13.0 km north. Use a graphical method to find the pedestrian\u2019s resultant displacement and geographic direction.<\/p>\n<\/div>\n<div id=\"fs-id1167132526037\">\n<p id=\"fs-id1167133862600\">14.3 km, [latex]65\\text{\u00b0}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133862614\">\n<div id=\"fs-id1167133862616\">\n<p id=\"fs-id1167133839798\">The magnitudes of two displacement vectors are <em>A<\/em> = 20 m and <em>B<\/em> = 6 m. What are the largest and the smallest values of the magnitude of the resultant [latex]\\stackrel{\\to }{R}=\\stackrel{\\to }{A}+\\stackrel{\\to }{B}?[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1167132688517\">\n<dt>antiparallel vectors<\/dt>\n<dd id=\"fs-id1167132688522\">two vectors with directions that differ by [latex]180\\text{\u00b0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579123\">\n<dt>associative<\/dt>\n<dd id=\"fs-id1167132579129\">terms can be grouped in any fashion<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579133\">\n<dt>commutative<\/dt>\n<dd id=\"fs-id1167132581795\">operations can be performed in any order<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132581799\">\n<dt>difference of two vectors<\/dt>\n<dd id=\"fs-id1167132581804\">vector sum of the first vector with the vector antiparallel to the second<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132581808\">\n<dt>displacement<\/dt>\n<dd id=\"fs-id1167133770991\">change in position<\/dd>\n<\/dl>\n<dl id=\"fs-id1167133770995\">\n<dt>distributive<\/dt>\n<dd id=\"fs-id1167133771000\">multiplication can be distributed over terms in summation<\/dd>\n<\/dl>\n<dl id=\"fs-id1167133771004\">\n<dt>magnitude<\/dt>\n<dd id=\"fs-id1167132504147\">length of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132504151\">\n<dt>orthogonal vectors<\/dt>\n<dd id=\"fs-id1167132504156\">two vectors with directions that differ by exactly [latex]90\\text{\u00b0}[\/latex], synonymous with perpendicular vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132707485\">\n<dt>parallelogram rule<\/dt>\n<dd id=\"fs-id1167132707491\">geometric construction of the vector sum in a plane<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132520196\">\n<dt>parallel vectors<\/dt>\n<dd id=\"fs-id1167132520201\">two vectors with exactly the same direction angles<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132520205\">\n<dt>resultant vector<\/dt>\n<dd id=\"fs-id1167132520210\">vector sum of two (or more) vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167128844182\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id1167128844188\">a number, synonymous with a scalar quantity in physics<\/dd>\n<\/dl>\n<dl id=\"fs-id1167128844192\">\n<dt>scalar equation<\/dt>\n<dd id=\"fs-id1167128844197\">equation in which the left-hand and right-hand sides are numbers<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132586350\">\n<dt>scalar quantity<\/dt>\n<dd id=\"fs-id1167132586355\">quantity that can be specified completely by a single number with an appropriate physical unit<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132586360\">\n<dt>tail-to-head geometric construction<\/dt>\n<dd id=\"fs-id1167132586366\">geometric construction for drawing the resultant vector of many vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132550483\">\n<dt>unit vector<\/dt>\n<dd id=\"fs-id1167132550488\">vector of a unit magnitude that specifies direction; has no physical unit<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132550493\">\n<dt>vector<\/dt>\n<dd id=\"fs-id1167132550498\">mathematical object with magnitude and direction<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579716\">\n<dt>vector equation<\/dt>\n<dd id=\"fs-id1167132579722\">equation in which the left-hand and right-hand sides are vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579726\">\n<dt>vector quantity<\/dt>\n<dd id=\"fs-id1167132579731\">physical quantity described by a mathematical vector\u2014that is, by specifying both its magnitude and its direction; synonymous with a vector in physics<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132445065\">\n<dt>vector sum<\/dt>\n<dd id=\"fs-id1167132445070\">resultant of the combination of two (or more) vectors<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Describe the difference between vector and scalar quantities.<\/li>\n<li>Identify the magnitude and direction of a vector.<\/li>\n<li>Explain the effect of multiplying a vector quantity by a scalar.<\/li>\n<li>Describe how one-dimensional vector quantities are added or subtracted.<\/li>\n<li>Explain the geometric construction for the addition or subtraction of vectors in a plane.<\/li>\n<li>Distinguish between a vector equation and a scalar equation.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167132379492\">Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, \u201ca class period lasts 50 min\u201d or \u201cthe gas tank in my car holds 65 L\u201d or \u201cthe distance between two posts is 100 m.\u201d A physical quantity that can be specified completely in this manner is called a <span>scalar quantity<\/span>. Scalar is a synonym of \u201cnumber.\u201d Time, mass, distance, length, volume, temperature, and energy are examples of <span>scalar<\/span> quantities.<\/p>\n<p id=\"fs-id1167133331747\">Scalar quantities that have the same physical units can be added or subtracted according to the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50 min lasts <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-50575c8e0bcb486a8d0bbe991d64a587_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#125;&#45;&#49;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#125;&#61;&#52;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"198\" style=\"vertical-align: -1px;\" \/>. Similarly, a 60-cal serving of corn followed by a 200-cal serving of donuts gives <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-96ee148db999e6ae2fe740dec06517a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#125;&#43;&#50;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#125;&#61;&#50;&#54;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"192\" style=\"vertical-align: -2px;\" \/> of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. For example, if yesterday\u2019s breakfast had 200 cal of energy and today\u2019s breakfast has four times as much energy as it had yesterday, then today\u2019s breakfast has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-51feec6872c3cd28cfab6631c7808ed7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\" \/> of energy. Two scalar quantities can also be multiplied or divided by each other to form a derived scalar quantity. For example, if a train covers a distance of 100 km in 1.0 h, its speed is 100.0 km\/1.0 h = 27.8 m\/s, where the speed is a derived scalar quantity obtained by dividing distance by time.<\/p>\n<p id=\"fs-id1167132683784\">Many physical quantities, however, cannot be described completely by just a single number of physical units. For example, when the U.S. Coast Guard dispatches a ship or a helicopter for a rescue mission, the rescue team must know not only the distance to the distress signal, but also the direction from which the signal is coming so they can get to its origin as quickly as possible. Physical quantities specified completely by giving a number of units (magnitude) and a direction are called <span>vector quantities<\/span>. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called <span>vectors<\/span> (<a href=\"#CNX_UPhysics_02_01_dog\" class=\"autogenerated-content\">(Figure)<\/a>). We can add or subtract two vectors, and we can multiply a vector by a scalar or by another vector, but we cannot divide by a vector. The operation of division by a vector is not defined.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_dog\">\n<div class=\"bc-figcaption figcaption\">We draw a vector from the initial point or origin (called the \u201ctail\u201d of a vector) to the end or terminal point (called the \u201chead\u201d of a vector), marked by an arrowhead. Magnitude is the length of a vector and is always a positive scalar quantity. (credit: modification of work by Cate Sevilla)<\/div>\n<p><span id=\"fs-id1167133854996\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_dog.jpg\" alt=\"A photo of a dog. Below the photo is a horizontal arrow which starts below the dog\u2019s tail and ends below the dog\u2019s nose. The arrow is labeled Vector D, and its length is labeled as magnitude D. The start (tail) of the arrow is labeled \u201cFrom rail of a vector origin\u201d and its end (head) is labeled \u201cTo head of a vector end.\u201d\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167133527625\">Let\u2019s examine vector algebra using a graphical method to be aware of basic terms and to develop a qualitative understanding. In practice, however, when it comes to solving physics problems, we use analytical methods, which we\u2019ll see in the next section. Analytical methods are more simple computationally and more accurate than graphical methods. From now on, to distinguish between a vector and a scalar quantity, we adopt the common convention that a letter in bold type with an arrow above it denotes a vector, and a letter without an arrow denotes a scalar. For example, a distance of 2.0 km, which is a scalar quantity, is denoted by <em>d<\/em> = 2.0 km, whereas a displacement of 2.0 km in some direction, which is a vector quantity, is denoted by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-07035500db9f51b2fe44d2c8867e226a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<p id=\"fs-id1167133333619\">Suppose you tell a friend on a camping trip that you have discovered a terrific fishing hole 6 km from your tent. It is unlikely your friend would be able to find the hole easily unless you also communicate the direction in which it can be found with respect to your campsite. You may say, for example, \u201cWalk about 6 km northeast from my tent.\u201d The key concept here is that you have to give not one but <em>two<\/em> pieces of information\u2014namely, the distance or magnitude (6 km) <em>and<\/em> the direction (northeast).<\/p>\n<p id=\"fs-id1167133802941\">Displacement is a general term used to describe a <em>change in position<\/em>, such as during a trip from the tent to the fishing hole. Displacement is an example of a vector quantity. If you walk from the tent (location <em>A<\/em>) to the hole (location <em>B<\/em>), as shown in <a href=\"#CNX_UPhysics_02_01_hole\" class=\"autogenerated-content\">(Figure)<\/a>, the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/>, representing your <span>displacement<\/span>, is drawn as the arrow that originates at point <em>A<\/em> and ends at point <em>B<\/em>. The arrowhead marks the end of the vector. The direction of the displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> is the direction of the arrow. The length of the arrow represents the <span>magnitude<\/span> <em>D<\/em> of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/>. Here, <em>D<\/em> = 6 km. Since the magnitude of a vector is its length, which is a positive number, the magnitude is also indicated by placing the absolute value notation around the symbol that denotes the vector; so, we can write equivalently that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-95b190ee23c88f66cbb6af1cced69cc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#92;&#101;&#113;&#117;&#105;&#118;&#32;&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"72\" style=\"vertical-align: -4px;\" \/>. To solve a vector problem graphically, we need to draw the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> to scale. For example, if we assume 1 unit of distance (1 km) is represented in the drawing by a line segment of length <em>u<\/em> = 2 cm, then the total displacement in this example is represented by a vector of length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-dcb1c5ff72bd48515fb8e3ca5d8b3f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#54;&#117;&#61;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"204\" style=\"vertical-align: -4px;\" \/>, as shown in <a href=\"#CNX_UPhysics_02_01_vector01\" class=\"autogenerated-content\">(Figure)<\/a>. Notice that here, to avoid confusion, we used <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7b5bdb51347bef1e784d849b9600aeac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"75\" style=\"vertical-align: 0px;\" \/> to denote the magnitude of the actual displacement and <em>d<\/em> = 12 cm to denote the length of its representation in the drawing.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_hole\">\n<div class=\"bc-figcaption figcaption\">The displacement vector from point <em>A<\/em> (the initial position at the campsite) to point <em>B<\/em> (the final position at the fishing hole) is indicated by an arrow with origin at point <em>A<\/em> and end at point <em>B<\/em>. The displacement is the same for any of the actual paths (dashed curves) that may be taken between points <em>A<\/em> and <em>B<\/em>.<\/div>\n<p><span id=\"fs-id1167133464547\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_hole.jpg\" alt=\"An illustration of a lake, some distance northeast from a tent. North is up on the page, east to the right. The tent is labeled as location A, and the lake as location B. A straight arrow starts at A and ends at B. Three meandering paths, shown as dashed lines, also start at A and end at B.\" \/><\/span><\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector01\">\n<div class=\"bc-figcaption figcaption\">A displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> of magnitude 6 km is drawn to scale as a vector of length 12 cm when the length of 2 cm represents 1 unit of displacement (which in this case is 1 km).<\/div>\n<p><span id=\"fs-id1167133318847\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector01.jpg\" alt=\"A ruler is shown, with distance measured in centimeters. A vector is shown as an arrow parallel to the ruler, extending from its end at 0 c m to 12 c m, and is labeled as vector D.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167133824259\">Suppose your friend walks from the campsite at <em>A<\/em> to the fishing pond at <em>B<\/em> and then walks back: from the fishing pond at <em>B<\/em> to the campsite at <em>A<\/em>. The magnitude of the displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> from <em>A<\/em> to <em>B<\/em> is the same as the magnitude of the displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c2032b59d7e9a46e802a0714d5148cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> from <em>B<\/em> to <em>A<\/em> (it equals 6 km in both cases), so we can write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6e224a507ef51d8a01668d6170a41f51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#123;&#68;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/>. However, vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> is <em>not<\/em> equal to vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c2032b59d7e9a46e802a0714d5148cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> because these two vectors have different directions: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-22c49cd5cf65c75476a4688e3e47fb64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#92;&#110;&#101;&#32;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"97\" style=\"vertical-align: -4px;\" \/>. In <a href=\"#CNX_UPhysics_02_01_hole\" class=\"autogenerated-content\">(Figure)<\/a>, vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c2032b59d7e9a46e802a0714d5148cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> would be represented by a vector with an origin at point <em>B<\/em> and an end at point <em>A<\/em>, indicating vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c2032b59d7e9a46e802a0714d5148cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> points to the southwest, which is exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-27bdda3969a0aa6f1afbe111d02e4073_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> opposite to the direction of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/>. We say that vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c2032b59d7e9a46e802a0714d5148cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> is <span>antiparallel<\/span> to vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> and write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-f0aa791987adc933548ff92400f1b284_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"97\" style=\"vertical-align: -3px;\" \/>, where the minus sign indicates the antiparallel direction.<\/p>\n<p id=\"fs-id1167133579247\">Two vectors that have identical directions are said to be <span>parallel vectors<\/span>\u2014meaning, they are <em>parallel<\/em> to each other. Two parallel vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> are equal, denoted by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5ba86dedcd081cfabdd58661023b4506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"42\" style=\"vertical-align: -2px;\" \/>, if and only if they have equal magnitudes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-2e6191c3f83ba2f81a68e0da276f8d30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#124;&#61;&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"87\" style=\"vertical-align: -4px;\" \/>. Two vectors with directions perpendicular to each other are said to be <span>orthogonal vectors<\/span>. These relations between vectors are illustrated in <a href=\"#CNX_UPhysics_02_01_vector02\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector02\">\n<div class=\"bc-figcaption figcaption\">Various relations between two vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>. (a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-77fbdf7dbee1a14ae6a44dfda1667797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#92;&#110;&#101;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"42\" style=\"vertical-align: -4px;\" \/> because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-49da06ea7a406e50b6a6b1f562fe6ca5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#92;&#110;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"51\" style=\"vertical-align: -4px;\" \/>. (b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-77fbdf7dbee1a14ae6a44dfda1667797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#92;&#110;&#101;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"42\" style=\"vertical-align: -4px;\" \/> because they are not parallel and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-49da06ea7a406e50b6a6b1f562fe6ca5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#92;&#110;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"51\" style=\"vertical-align: -4px;\" \/>. (c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4f645a47c0102cf0a76152f462bc259e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#92;&#110;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"51\" style=\"vertical-align: -4px;\" \/> because they have different directions (even though <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-68b504b734215344982ca10c89fd44c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#124;&#61;&#124;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#124;&#61;&#65;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"139\" style=\"vertical-align: -4px;\" \/>. (d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5ba86dedcd081cfabdd58661023b4506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"42\" style=\"vertical-align: -2px;\" \/> because they are parallel <em>and<\/em> have identical magnitudes <em>A<\/em> = <em>B.<\/em> (e) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-77fbdf7dbee1a14ae6a44dfda1667797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#92;&#110;&#101;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"42\" style=\"vertical-align: -4px;\" \/> because they have different directions (are not parallel); here, their directions differ by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ad7fa5a1576b957af1a577f736d676e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u2014meaning, they are orthogonal.<\/div>\n<p><span id=\"fs-id1167132332563\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector02.jpg\" alt=\"Figure a: Two examples of vector A parallel to vector B. In one, A and B are on the same line, one after the other, but A is longer than B. In the other, A and B are parallel to each other with their tails aligned, but A is shorter than B. Figure b: An example of vector A antiparallel to vector B. Vector A points to the left and is longer than vector B, which points to the right. The angle between them is 180 degrees. Figure c: An example of vector A antiparallel to minus vector A: A points to the right and \u2013A points to the left. Both are the same length. Figure d: Two examples of vector A equal to vector B: In one, A and B are on the same line, one after the other, and both are the same length. In the other, A and B are parallel to each other with their tails aligned, and both are the same length. Figure e: Two examples of vector A orthogonal to vector B: In one, A points down and B points to the right, meeting at a right angle, and both are the same length. In the other, points down and to the right and B points down and to the left, meeting A at a right angle. Both are the same length.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1167132576708\" class=\"check-understanding\">\n<div id=\"fs-id1167132485087\">\n<div id=\"fs-id1167132240816\">\n<p id=\"fs-id1167132539326\"><strong>Check Your Understanding<\/strong> Two motorboats named <em>Alice<\/em> and <em>Bob<\/em> are moving on a lake. Given the information about their velocity vectors in each of the following situations, indicate whether their velocity vectors are equal or otherwise. (a) <em>Alice<\/em> moves north at 6 knots and <em>Bob<\/em> moves west at 6 knots. (b) <em>Alice<\/em> moves west at 6 knots and <em>Bob<\/em> moves west at 3 knots. (c) <em>Alice<\/em> moves northeast at 6 knots and <em>Bob<\/em> moves south at 3 knots. (d) <em>Alice<\/em> moves northeast at 6 knots and <em>Bob<\/em> moves southwest at 6 knots. (e) <em>Alice<\/em> moves northeast at 2 knots and <em>Bob<\/em> moves closer to the shore northeast at 2 knots.<\/p>\n<\/div>\n<div id=\"fs-id1167132519498\">\n<p id=\"fs-id1167132595446\">a. not equal because they are orthogonal; b. not equal because they have different magnitudes; c. not equal because they have different magnitudes and directions; d. not equal because they are antiparallel; e. equal.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1167132305565\">\n<h3>Algebra of Vectors in One Dimension<\/h3>\n<p id=\"fs-id1167132274736\">Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. We can illustrate these vector concepts using an example of the fishing trip seen in <a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_FishTrip\">\n<div class=\"bc-figcaption figcaption\">Displacement vectors for a fishing trip. (a) Stopping to rest at point <em>C<\/em> while walking from camp (point <em>A<\/em>) to the pond (point <em>B<\/em>). (b) Going back for the dropped tackle box (point <em>D<\/em>). (c) Finishing up at the fishing pond.<\/div>\n<p><span id=\"fs-id1167133792347\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_FishTrip.jpg\" alt=\"Three illustrations of the same tent and lake northeast of the tent. North is up on the page. The tent location is point A, and the lake location is point B. A location between A and B, about 2\/3 of the way from A to B, is labeled as point C. In figure a, the vector from A to B is shown as a blue arrow, starting at A and ending at B, and labeled vector D sub A B. The vector from A to C is shown as a red arrow, starting at A and ending at C and labeled vector D sub A C. Three meandering paths are shown as dashed lines that start at A and end at B. Figure b adds the following to the illustration of figure a: Point D is added about half way between point A and B. The vector from A to D is shown as a purple arrow, starting at A and ending at D and labeled vector D sub A D. The vector from D to B is shown as an orange arrow, starting at D and ending at B and labeled vector D sub D B. Figure c adds a green arrow from point C to point D and is labeled vector D sub C D. Vector D sub C D points in the direction opposite to that of the other vectors, toward the tent rather than toward the lake.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167133328285\">Suppose your friend departs from point <em>A<\/em> (the campsite) and walks in the direction to point <em>B<\/em> (the fishing pond), but, along the way, stops to rest at some point <em>C<\/em> located three-quarters of the distance between <em>A<\/em> and <em>B<\/em>, beginning from point <em>A<\/em> (<a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>(a)). What is his displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-94433f39432ca035a96d382f36b9ad6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> when he reaches point <em>C<\/em>? We know that if he walks all the way to <em>B<\/em>, his displacement vector relative to <em>A<\/em> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/>, which has magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-835bf2df142bf832cf874058c48a6f3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -3px;\" \/> and a direction of northeast. If he walks only a 0.75 fraction of the total distance, maintaining the northeasterly direction, at point <em>C<\/em> he must be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-1d22a0ac46e14202a966269be642c2c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#55;&#53;&#123;&#68;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#52;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"142\" style=\"vertical-align: -3px;\" \/> away from the campsite at <em>A<\/em>. So, his displacement vector at the rest point <em>C<\/em> has magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b4b27f8218964b999c4648e1479c55f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#65;&#67;&#125;&#61;&#52;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;&#61;&#48;&#46;&#55;&#53;&#123;&#68;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"202\" style=\"vertical-align: -3px;\" \/> and is parallel to the displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/>. All of this can be stated succinctly in the form of the following <span>vector equation<\/span>:<\/p>\n<div id=\"fs-id1167132497239\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-9ce9cc0a01ad6f4e425e4eb02824f05d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#67;&#125;&#61;&#48;&#46;&#55;&#53;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"133\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"fs-id1167132624842\">In a vector equation, both sides of the equation are vectors. The previous equation is an example of a vector multiplied by a positive scalar (number) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e9fedc2484d4d56f7a9be52ee392dc5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#48;&#46;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: 0px;\" \/>. The result, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-94433f39432ca035a96d382f36b9ad6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/>, of such a multiplication is a new vector with a direction parallel to the direction of the original vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p id=\"fs-id1167133675760\">In general, when a vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> is multiplied by a <em>positive<\/em> scalar <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, the result is a new vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> that is <em>parallel<\/em> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>:<\/p>\n<div id=\"fs-id1167132262787\" class=\"equation-callout\">\n<div id=\"fs-id1167133686173\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-aaa08f59caa0eeb5aa5d668aaccddede_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#61;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"72\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167133750753\">The magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d77549b68f8aecb4ba202780df63473e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"30\" style=\"vertical-align: -4px;\" \/> of this new vector is obtained by multiplying the magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-cad1a25e9f8e725390f4acd76d12e967_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"30\" style=\"vertical-align: -4px;\" \/> of the original vector, as expressed by the <span>scalar equation<\/span>:<\/p>\n<div id=\"fs-id1167132320492\" class=\"equation-callout\">\n<div id=\"fs-id1167133394048\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-da1511d61e1aa04d6630b2f2750ee599_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#61;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#124;&#65;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167132413817\">In a scalar equation, both sides of the equation are numbers. <a href=\"#fs-id1167133394048\" class=\"autogenerated-content\">(Figure)<\/a> is a scalar equation because the magnitudes of vectors are scalar quantities (and positive numbers). If the scalar <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is <em>negative<\/em> in the vector equation <a href=\"#fs-id1167133686173\" class=\"autogenerated-content\">(Figure)<\/a>, then the magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d77549b68f8aecb4ba202780df63473e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"30\" style=\"vertical-align: -4px;\" \/> of the new vector is still given by <a href=\"#fs-id1167133394048\" class=\"autogenerated-content\">(Figure)<\/a>, but the direction of the new vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> is <em>antiparallel<\/em> to the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>. These principles are illustrated in <a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>(a) by two examples where the length of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> is 1.5 units. When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-95de397a84e01f53b63dd49d97858ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>, the new vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-9ae8a313e9db67f2d3782270c54a05d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#61;&#50;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"60\" style=\"vertical-align: -2px;\" \/> has length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ad2e639a512eb41ba8053057dbb29768_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#61;&#50;&#65;&#61;&#51;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#110;&#105;&#116;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"149\" style=\"vertical-align: 0px;\" \/> (twice as long as the original vector) and is parallel to the original vector. When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4c4430ae5cbd243761a9fc32c9d38eb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/>, the new vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a765f77d494dde08a7258f4adccff0ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;&#61;&#45;&#50;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"73\" style=\"vertical-align: -2px;\" \/> has length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a5b3ec48f3663c623eee5afceb75bb63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#124;&#45;&#50;&#124;&#65;&#61;&#51;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#110;&#105;&#116;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"180\" style=\"vertical-align: -4px;\" \/> (twice as long as the original vector) and is antiparallel to the original vector.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector03\">\n<div class=\"bc-figcaption figcaption\">Algebra of vectors in one dimension. (a) Multiplication by a scalar. (b) Addition of two vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-f97824e5d42753a8897f24c47b21c858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"22\" style=\"vertical-align: -12px;\" \/> is called the <em>resultant<\/em> of vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d0342373f422bc7a6817ff6920a48f5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>. (c) Subtraction of two vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e8e471825ea080d10ef1bf3478f3754a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"23\" style=\"vertical-align: -12px;\" \/> is the difference of vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d0342373f422bc7a6817ff6920a48f5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>.<\/div>\n<p><span id=\"fs-id1167132221972\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector03.jpg\" alt=\"Figure a shows vector A pointing to the right. It has magnitude A=1.5. Vector B=2 time vector A points to the right and has magnitude B = 2 A = 3.0. Vector C = -2 times vector A and has magnitude B = 2.0. Figure b shows vector A points to the right and has magnitude A=1.5. Vector B is shown below vector A, with their tails aligned. Vector B points to the right and has magnitude 2.0. In another view, Vector A is shown with vector B starting at the head of A and extending further to the right. Below them is a vector, labeled as vector R = vector A plus vector B, pointing to the right whose tail is aligned with the tail of vector A and whose head is aligned with the head of vector B. The magnitude of vector R is equal to magnitude A plus magnitude B = 3.5. Figure c shows vector A points to the right and has magnitude A=1.5. Vector B is shown below vector A, with their tails aligned. Vector minus B points to the right and has magnitude 3.2. In another view, Vector A is shown with vector minus B pointing to the left and with its head meeting the head of vector A. Below them is a vector, labeled as vector D = vector A minus vector B, shorter than B and pointing to the left whose head is aligned with the head of vector B. The magnitude of vector D is equal to magnitude of quantity A minus B = 1.7.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167133579584\">Now suppose your fishing buddy departs from point <em>A<\/em> (the campsite), walking in the direction to point <em>B<\/em> (the fishing hole), but he realizes he lost his tackle box when he stopped to rest at point <em>C<\/em> (located three-quarters of the distance between <em>A<\/em> and <em>B<\/em>, beginning from point <em>A<\/em>). So, he turns back and retraces his steps in the direction toward the campsite and finds the box lying on the path at some point <em>D<\/em> only 1.2 km away from point <em>C<\/em> (see <a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>(b)). What is his displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d9ba0da38150d5f371863953d890d316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> when he finds the box at point <em>D<\/em>? What is his displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d4461e5d2ba38c6e27c84b4618ea7a4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#68;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"38\" style=\"vertical-align: -3px;\" \/> from point <em>D<\/em> to the hole? We have already established that at rest point <em>C<\/em> his displacement vector is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-614bed4c8cce4bd9a23b8bff8aa7a745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#67;&#125;&#61;&#48;&#46;&#55;&#53;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"128\" style=\"vertical-align: -3px;\" \/>. Starting at point <em>C<\/em>, he walks southwest (toward the campsite), which means his new displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b229bd412d553cd2d9693b13d0d70ee1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#67;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"37\" style=\"vertical-align: -3px;\" \/> from point <em>C<\/em> to point <em>D<\/em> is antiparallel to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/>. Its magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b84f76c6633e6ed10f478757a51a5e88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#67;&#68;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"44\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-86177d540a21f74565a0202c4d642ac1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#67;&#68;&#125;&#61;&#49;&#46;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;&#61;&#48;&#46;&#50;&#123;&#68;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"194\" style=\"vertical-align: -3px;\" \/>, so his second displacement vector is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5c04c390ea8325d3285033da737f6efb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#67;&#68;&#125;&#61;&#45;&#48;&#46;&#50;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"135\" style=\"vertical-align: -3px;\" \/>. His total displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d9ba0da38150d5f371863953d890d316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> relative to the campsite is the <span>vector sum<\/span> of the two displacement vectors: vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-94433f39432ca035a96d382f36b9ad6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> (from the campsite to the rest point) and vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b229bd412d553cd2d9693b13d0d70ee1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#67;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"37\" style=\"vertical-align: -3px;\" \/> (from the rest point to the point where he finds his box):<\/p>\n<div id=\"fs-id1167132225666\" class=\"equation-callout\">\n<div id=\"fs-id1167132476572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-907aee1eb52095cb3f9a65f108e29a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#67;&#125;&#43;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#67;&#68;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"163\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167132507760\">The vector sum of two (or more) vectors is called the <span>resultant vector<\/span> or, for short, the <em>resultant<\/em>. When the vectors on the right-hand-side of <a href=\"#fs-id1167132476572\" class=\"autogenerated-content\">(Figure)<\/a> are known, we can find the resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d9ba0da38150d5f371863953d890d316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> as follows:<\/p>\n<div id=\"fs-id1167133527385\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c90e55a2735838105a11be090d60dab0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#67;&#125;&#43;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#67;&#68;&#125;&#61;&#48;&#46;&#55;&#53;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#45;&#48;&#46;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#55;&#53;&#45;&#48;&#46;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#48;&#46;&#53;&#53;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"586\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167132416496\">When your friend finally reaches the pond at <em>B<\/em>, his displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> from point <em>A<\/em> is the vector sum of his displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d9ba0da38150d5f371863953d890d316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> from point <em>A<\/em> to point <em>D<\/em> and his displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d4461e5d2ba38c6e27c84b4618ea7a4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#68;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"38\" style=\"vertical-align: -3px;\" \/> from point <em>D<\/em> to the fishing hole: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-246545e87bc10a3a0215deae70838e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;&#43;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#68;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"158\" style=\"vertical-align: -3px;\" \/> (see <a href=\"#CNX_UPhysics_02_01_FishTrip\" class=\"autogenerated-content\">(Figure)<\/a>(c)). This means his displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d4461e5d2ba38c6e27c84b4618ea7a4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#68;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"38\" style=\"vertical-align: -3px;\" \/> is the <span>difference of two vectors<\/span>:<\/p>\n<div id=\"fs-id1167132441548\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-759ff85e2adf32ada828eaf160c8ed6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#68;&#66;&#125;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#45;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"308\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"fs-id1167132281683\">Notice that a difference of two vectors is nothing more than a vector sum of two vectors because the second term in <a href=\"#fs-id1167132441548\" class=\"autogenerated-content\">(Figure)<\/a> is vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8ab018275dc56c868bb4579dc36cd110_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> (which is antiparallel to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c1ec6cc72256f4720cdc8448022b9b02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/>. When we substitute <a href=\"#fs-id1167133527385\" class=\"autogenerated-content\">(Figure)<\/a> into <a href=\"#fs-id1167132441548\" class=\"autogenerated-content\">(Figure)<\/a>, we obtain the second displacement vector:<\/p>\n<div id=\"fs-id1167133768313\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-032e94edc055545079b88f9a0095a808_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#68;&#66;&#125;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#45;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#68;&#125;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#45;&#48;&#46;&#53;&#53;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#46;&#48;&#45;&#48;&#46;&#53;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#48;&#46;&#52;&#53;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"561\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167132499982\">This result means your friend walked <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ba3b6d04c168f5f00e20cf850f214326_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#68;&#66;&#125;&#61;&#48;&#46;&#52;&#53;&#123;&#68;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#48;&#46;&#52;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#46;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"326\" style=\"vertical-align: -4px;\" \/> from the point where he finds his tackle box to the fishing hole.<\/p>\n<p id=\"fs-id1167132237953\">When vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> lie along a line (that is, in one dimension), such as in the camping example, their resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ffe1f1647343ce3bedcdbee06e7ac084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -2px;\" \/> and their difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4fd74968ec6116e483a6cea0e8fabec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -2px;\" \/> both lie along the same direction. We can illustrate the addition or subtraction of vectors by drawing the corresponding vectors to scale in one dimension, as shown in <a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1167132378007\">To illustrate the resultant when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> are two parallel vectors, we draw them along one line by placing the origin of one vector at the end of the other vector in head-to-tail fashion (see <a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>(b)). The magnitude of this resultant is the sum of their magnitudes: <em>R<\/em> = <em>A<\/em> + <em>B<\/em>. The direction of the resultant is parallel to both vectors. When vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> is antiparallel to vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, we draw them along one line in either head-to-head fashion (<a href=\"#CNX_UPhysics_02_01_vector03\" class=\"autogenerated-content\">(Figure)<\/a>(c)) or tail-to-tail fashion. The magnitude of the vector difference, then, is the <em>absolute value<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c04ec2730f7a0409b9dd90d7dabfcba5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#124;&#65;&#45;&#66;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/> of the difference of their magnitudes. The direction of the difference vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> is parallel to the direction of the longer vector.<\/p>\n<p id=\"fs-id1167132299184\">In general, in one dimension\u2014as well as in higher dimensions, such as in a plane or in space\u2014we can add any number of vectors and we can do so in any order because the addition of vectors is <span>commutative<\/span>,<\/p>\n<div id=\"fs-id1167132463322\" class=\"equation-callout\">\n<div id=\"fs-id1167132501766\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-faeb0c323692af20dac945e63e7dbb7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167132569455\">and <span>associative<\/span>,<\/p>\n<div id=\"fs-id1167132239544\" class=\"equation-callout\">\n<div id=\"fs-id1167133454633\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c2ee42b35eeb53f4863e05d5cc82375f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"235\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167132342483\">Moreover, multiplication by a scalar is <span>distributive<\/span>:<\/p>\n<div id=\"fs-id1167132615617\" class=\"equation-callout\">\n<div id=\"fs-id1167132615620\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-dccd5eff9f094dbfeb1bc973f348aaf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;&#95;&#123;&#50;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"214\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167132413323\">We used the distributive property in <a href=\"#fs-id1167133527385\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1167133768313\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1167132397556\">When adding many vectors in one dimension, it is convenient to use the concept of a <span>unit vector<\/span>. A unit vector, which is denoted by a letter symbol with a hat, such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-08dc3a6513d93ffc281cd2740aa93e16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 1px;\" \/>, has a magnitude of one and does not have any physical unit so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-833e68446ab952988e895f4a5e27394a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#124;&#92;&#101;&#113;&#117;&#105;&#118;&#32;&#117;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -4px;\" \/>. The only role of a unit vector is to specify direction. For example, instead of saying vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de3dc014e7c801782ed4656cb04e7801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -3px;\" \/> has a magnitude of 6.0 km and a direction of northeast, we can introduce a unit vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-08dc3a6513d93ffc281cd2740aa93e16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 1px;\" \/> that points to the northeast and say succinctly that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-2834563bcc3162945dcf01d5340d9ef2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"140\" style=\"vertical-align: -4px;\" \/>. Then the southwesterly direction is simply given by the unit vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-30c7c2ee9a6715572df0627564f3bc10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 1px;\" \/>. In this way, the displacement of 6.0 km in the southwesterly direction is expressed by the vector<\/p>\n<div id=\"fs-id1167132499315\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-05e7c1737e5c1d00b014e903bf473d81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#66;&#65;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/div>\n<div id=\"fs-id1167132203045\" class=\"textbox examples\">\n<p id=\"fs-id1167132203047\"><span>A Ladybug Walker<\/span><br \/>\nA long measuring stick rests against a wall in a physics laboratory with its 200-cm end at the floor. A ladybug lands on the 100-cm mark and crawls randomly along the stick. It first walks 15 cm toward the floor, then it walks 56 cm toward the wall, then it walks 3 cm toward the floor again. Then, after a brief stop, it continues for 25 cm toward the floor and then, again, it crawls up 19 cm toward the wall before coming to a complete rest (<a href=\"#CNX_UPhysics_02_01_ladybug\" class=\"autogenerated-content\">(Figure)<\/a>). Find the vector of its total displacement and its final resting position on the stick.<\/p>\n<p id=\"fs-id1167132688912\"><span>Strategy<\/span><br \/>\nIf we choose the direction along the stick toward the floor as the direction of unit vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-08dc3a6513d93ffc281cd2740aa93e16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 1px;\" \/>, then the direction toward the floor is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-225afdc84626c1c9a6ca620a5aa229e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"29\" style=\"vertical-align: -2px;\" \/> and the direction toward the wall is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-30c7c2ee9a6715572df0627564f3bc10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 1px;\" \/>. The ladybug makes a total of five displacements:<\/p>\n<div id=\"fs-id1167133508498\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7a48b5f62af4ebe5f6096b10b3b25fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#49;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#51;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#52;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#53;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"162\" width=\"201\" style=\"vertical-align: -77px;\" \/><\/div>\n<p id=\"fs-id1167132241640\">The total displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> is the resultant of all its displacement vectors.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_ladybug\">\n<div class=\"bc-figcaption figcaption\">Five displacements of the ladybug. Note that in this schematic drawing, magnitudes of displacements are not drawn to scale. (credit: modification of work by \u201cPersian Poet Gal\u201d\/Wikimedia Commons)<\/div>\n<p><span id=\"fs-id1167133748196\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_ladybug.jpg\" alt=\"Five illustrations of a ladybug on a ruler leaning against a wall. The +u hat direction is toward the floor parallel to the ruler, and the \u2013 u hat direction is up along the ruler. In the first illustration, the ladybug is located near the middle of the ruler and vector D sub 1 points down the ruler. In the second illustration, the ladybug is located lower, where the head of vector D sub 1 is in the first illustration, and vector D sub 2 points up the ruler. In the third illustration, the ladybug is located higher, where the head of vector D sub 2 is in the second illustration, and vector D sub 3 points down the ruler. In the fourth illustration, the ladybug is located lower, where the head of vector D sub 3 is in the third illustration, and vector D sub 4 points down the ruler. In the fifth illustration, the ladybug is located lower, where the head of vector D sub 4 is in the fourth illustration, and vector D sub 5 points up the ruler.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167133826776\"><span>Solution<\/span><br \/>\nThe resultant of all the displacement vectors is<\/p>\n<div id=\"fs-id1167133826781\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5b390fc37ae0460d97d689677e5f18e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#38;&#32;&#61;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#51;&#125;&#43;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#52;&#125;&#43;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#125;&#95;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#45;&#53;&#54;&#43;&#51;&#43;&#50;&#53;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#61;&#45;&#51;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"651\" style=\"vertical-align: -43px;\" \/><\/div>\n<p id=\"fs-id1167132436133\">In this calculation, we use the distributive law given by <a href=\"#fs-id1167132615620\" class=\"autogenerated-content\">(Figure)<\/a>. The result reads that the total displacement vector points away from the 100-cm mark (initial landing site) toward the end of the meter stick that touches the wall. The end that touches the wall is marked 0 cm, so the final position of the ladybug is at the (100 \u2013 32)cm = 68-cm mark.<\/p>\n<\/div>\n<div id=\"fs-id1167133521609\" class=\"check-understanding\">\n<div id=\"fs-id1167132473641\">\n<div id=\"fs-id1167132473643\">\n<p id=\"fs-id1167132473645\"><strong>Check Your Understanding<\/strong> A cave diver enters a long underwater tunnel. When her displacement with respect to the entry point is 20 m, she accidentally drops her camera, but she doesn\u2019t notice it missing until she is some 6 m farther into the tunnel. She swims back 10 m but cannot find the camera, so she decides to end the dive. How far from the entry point is she? Taking the positive direction out of the tunnel, what is her displacement vector relative to the entry point?<\/p>\n<\/div>\n<div id=\"fs-id1167133773324\">\n<p id=\"fs-id1167133773326\">16 m; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5d1375d9e82f9a203e565e506e3aa619_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#61;&#45;&#49;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"99\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1167132371115\">\n<h3>Algebra of Vectors in Two Dimensions<\/h3>\n<p id=\"fs-id1167132342179\">When vectors lie in a plane\u2014that is, when they are in two dimensions\u2014they can be multiplied by scalars, added to other vectors, or subtracted from other vectors in accordance with the general laws expressed by <a href=\"#fs-id1167133686173\" class=\"autogenerated-content\">(Figure)<\/a>, <a href=\"#fs-id1167133394048\" class=\"autogenerated-content\">(Figure)<\/a>, <a href=\"#fs-id1167132501766\" class=\"autogenerated-content\">(Figure)<\/a>, and <a href=\"#fs-id1167133454633\" class=\"autogenerated-content\">(Figure)<\/a>. However, the addition rule for two vectors in a plane becomes more complicated than the rule for vector addition in one dimension. We have to use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. This geometric approach is commonly used in navigation (<a href=\"#CNX_UPhysics_02_01_navigation\" class=\"autogenerated-content\">(Figure)<\/a>). In this section, we need to have at hand two rulers, a triangle, a protractor, a pencil, and an eraser for drawing vectors to scale by geometric constructions.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_navigation\">\n<div class=\"bc-figcaption figcaption\">In navigation, the laws of geometry are used to draw resultant displacements on nautical maps.<\/div>\n<p><span id=\"fs-id1167133541239\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_navigation.jpg\" alt=\"A photograph of someone measuring distance on a map using calipers and a ruler.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167132594682\">For a geometric construction of the sum of two vectors in a plane, we follow <span>the parallelogram rule<\/span>. Suppose two vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> are at the arbitrary positions shown in <a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>. Translate either one of them in parallel to the beginning of the other vector, so that after the translation, both vectors have their origins at the same point. Now, at the end of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> we draw a line parallel to vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> and at the end of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> we draw a line parallel to vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> (the dashed lines in <a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>). In this way, we obtain a parallelogram. From the origin of the two vectors we draw a diagonal that is the resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6bb3969d03d2d1cf2c9f16860552e6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> of the two vectors: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ffe1f1647343ce3bedcdbee06e7ac084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -2px;\" \/> (<a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>(a)). The other diagonal of this parallelogram is the vector difference of the two vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4fd74968ec6116e483a6cea0e8fabec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -2px;\" \/>, as shown in <a href=\"#CNX_UPhysics_02_01_vector04\" class=\"autogenerated-content\">(Figure)<\/a>(b). Notice that the end of the difference vector is placed at the end of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector04\">\n<div class=\"bc-figcaption figcaption\">The parallelogram rule for the addition of two vectors. Make the parallel translation of each vector to a point where their origins (marked by the dot) coincide and construct a parallelogram with two sides on the vectors and the other two sides (indicated by dashed lines) parallel to the vectors. (a) Draw the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6bb3969d03d2d1cf2c9f16860552e6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> along the diagonal of the parallelogram from the common point to the opposite corner. Length <em>R<\/em> of the resultant vector is <em>not<\/em> equal to the sum of the magnitudes of the two vectors. (b) Draw the difference vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4fd74968ec6116e483a6cea0e8fabec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -2px;\" \/> along the diagonal connecting the ends of the vectors. Place the origin of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> at the end of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> and the end (arrowhead) of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> at the end of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>. Length <em>D<\/em> of the difference vector is <em>not<\/em> equal to the difference of magnitudes of the two vectors.<\/div>\n<p><span id=\"fs-id1167133635872\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector04.jpg\" alt=\"The parallelogram method for adding vectors is illustrated. In figure a, vectors A and B are shown. Vector A points to the right and down and vector B points right and up. Vectors A and B are then shown as solid arrows with their tails together, and their directions as before. A dashed line parallel to vector A but shifted so it starts at the head of B is shown. A second dashed line, parallel to B and starting at the head of A is also shown. The vectors A and B and the two dashed lines form a parallelogram. A third vector, labeled vector R = vector A plus vector B, is shown. The tail of vector R is at the tails of vectors A and B, and the head of vector R is where the dashed lines meet each other, diagonally across the parallelogram. We note that the magnitude of R is not equal to the magnitude of A plus the magnitude of B. In figure b, vectors A and minus B are shown. Vector minus B is vector B from part a, rotated 180 degrees. Vector A points to the right and down and vector minus B points left and down. Vectors A and B are then shown as solid arrows with their tails together, and their directions as before. A dashed line parallel to vector A but shifted so it starts at the head of B is shown. A second dashed line, parallel to B and starting at the head of A is also shown. The vectors A and B and the two dashed lines form a parallelogram. A third vector, labeled vector D is shown. The tail of vector D is at the head of vector B, and the head of vector D is at the head of vector A, diagonally across the parallelogram. We note that vector D is equal to vector A minus vector B, but the magnitude of D is not equal to the magnitude of A minus the B.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167132362973\">It follows from the parallelogram rule that neither the magnitude of the resultant vector nor the magnitude of the difference vector can be expressed as a simple sum or difference of magnitudes <em>A<\/em> and <em>B<\/em>, because the length of a diagonal cannot be expressed as a simple sum of side lengths. When using a geometric construction to find magnitudes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-68ba839b80d2761b9720743bc1784c24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"30\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-08b6abcae6cdf2da2594addc8fba147a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"31\" style=\"vertical-align: -4px;\" \/>, we have to use trigonometry laws for triangles, which may lead to complicated algebra. There are two ways to circumvent this algebraic complexity. One way is to use the method of components, which we examine in the next section. The other way is to draw the vectors to scale, as is done in navigation, and read approximate vector lengths and angles (directions) from the graphs. In this section we examine the second approach.<\/p>\n<p id=\"fs-id1167132501642\">If we need to add three or more vectors, we repeat the parallelogram rule for the pairs of vectors until we find the resultant of all of the resultants. For three vectors, for example, we first find the resultant of vector 1 and vector 2, and then we find the resultant of this resultant and vector 3. The order in which we select the pairs of vectors does not matter because the operation of vector addition is commutative and associative (see <a href=\"#fs-id1167132501766\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1167133454633\" class=\"autogenerated-content\">(Figure)<\/a>). Before we state a general rule that follows from repetitive applications of the parallelogram rule, let\u2019s look at the following example.<\/p>\n<p id=\"fs-id1167132267118\">Suppose you plan a vacation trip in Florida. Departing from Tallahassee, the state capital, you plan to visit your uncle Joe in Jacksonville, see your cousin Vinny in Daytona Beach, stop for a little fun in Orlando, see a circus performance in Tampa, and visit the University of Florida in Gainesville. Your route may be represented by five displacement vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5cff696cbc5c6ff4e822202bc51ba77f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"18\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bef3089b70eb979b319560cdb4c43c4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-45bee663123a81e2f73ca496d37404ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#69;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, which are indicated by the red vectors in <a href=\"#CNX_UPhysics_02_01_vector05\" class=\"autogenerated-content\">(Figure)<\/a>. What is your total displacement when you reach Gainesville? The total displacement is the vector sum of all five displacement vectors, which may be found by using the parallelogram rule four times. Alternatively, recall that the displacement vector has its beginning at the initial position (Tallahassee) and its end at the final position (Gainesville), so the total displacement vector can be drawn directly as an arrow connecting Tallahassee with Gainesville (see the green vector in <a href=\"#CNX_UPhysics_02_01_vector05\" class=\"autogenerated-content\">(Figure)<\/a>). When we use the parallelogram rule four times, the resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6bb3969d03d2d1cf2c9f16860552e6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> we obtain is exactly this green vector connecting Tallahassee with Gainesville: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-1b6b20fddc0b5a9a07aa38b97640fb67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#69;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"193\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector05\">\n<div class=\"bc-figcaption figcaption\">When we use the parallelogram rule four times, we obtain the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-1b6b20fddc0b5a9a07aa38b97640fb67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#69;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"193\" style=\"vertical-align: -2px;\" \/>, which is the green vector connecting Tallahassee with Gainesville.<\/div>\n<p><span id=\"fs-id1167132322121\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector05.jpg\" alt=\"A map of Florida with the following vectors shown in red: Vector A from Tallahassee to Jacksonville, almost due west. Vector B from Jacksonville to Daytona Beach, southeast. Vector C from Daytona Beach to Orlando, southwest. Vector D from Orlando to Tampa, southwest (but less vertical than vector C). Vector E from Tampa to Gainesville, slightly east of north. Vector R from Tallahassee to Gainsville is shown as a green arrow.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167132444404\">Drawing the resultant vector of many vectors can be generalized by using the following <span>tail-to-head geometric construction<\/span>. Suppose we want to draw the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6bb3969d03d2d1cf2c9f16860552e6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> of four vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bef3089b70eb979b319560cdb4c43c4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/> (<a href=\"#CNX_UPhysics_02_01_vector06\" class=\"autogenerated-content\">(Figure)<\/a>(a)). We select any one of the vectors as the first vector and make a parallel translation of a second vector to a position where the origin (\u201ctail\u201d) of the second vector coincides with the end (\u201chead\u201d) of the first vector. Then, we select a third vector and make a parallel translation of the third vector to a position where the origin of the third vector coincides with the end of the second vector. We repeat this procedure until all the vectors are in a head-to-tail arrangement like the one shown in <a href=\"#CNX_UPhysics_02_01_vector06\" class=\"autogenerated-content\">(Figure)<\/a>. We draw the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6bb3969d03d2d1cf2c9f16860552e6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> by connecting the origin (\u201ctail\u201d) of the first vector with the end (\u201chead\u201d) of the last vector. The end of the resultant vector is at the end of the last vector. Because the addition of vectors is associative and commutative, we obtain the same resultant vector regardless of which vector we choose to be first, second, third, or fourth in this construction.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector06\">\n<div class=\"bc-figcaption figcaption\">Tail-to-head method for drawing the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7e2690df4ad22edad5467791e0266179_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"155\" style=\"vertical-align: -2px;\" \/>. (a) Four vectors of different magnitudes and directions. (b) Vectors in (a) are translated to new positions where the origin (\u201ctail\u201d) of one vector is at the end (\u201chead\u201d) of another vector. The resultant vector is drawn from the origin (\u201ctail\u201d) of the first vector to the end (\u201chead\u201d) of the last vector in this arrangement.<\/div>\n<p><span id=\"fs-id1167132205226\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector06.jpg\" alt=\"In figure a, four vectors, labeled A, B, C, and D are shown individually. In figure b, the vectors are shown arranged head to tail: Vector A\u2019s tail is at the head of D. Vector C\u2019s tail is at the head of A. And vector B\u2019s tail is at the head of C. Each vector is pointing in the same direction as it is in figure a. A fifth vector, R, starts at the tail of vector D and ends at the head of vector B.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1167132310249\" class=\"textbox examples\">\n<p id=\"fs-id1167132546290\"><span>Geometric Construction of the Resultant<\/span><br \/>\nThe three displacement vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bef3089b70eb979b319560cdb4c43c4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> in <a href=\"#CNX_UPhysics_02_01_vector07\" class=\"autogenerated-content\">(Figure)<\/a> are specified by their magnitudes <em>A<\/em> = 10.0, <em>B<\/em> = 7.0, and <em>C<\/em> = 8.0, respectively, and by their respective direction angles with the horizontal direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6b66ca15dff9fa1f614e6ed84f608127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#51;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c2852ee1ba4ddd03cd6a1e8e2ced17dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#45;&#49;&#49;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -4px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d52b3c8de80dee97902a1dc6c67be0e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#51;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/>. The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ffe1f1647343ce3bedcdbee06e7ac084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -2px;\" \/>, (b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-f0a7054f7c2808afc6e3289e899da44a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#32;&#97;&#110;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"125\" style=\"vertical-align: -3px;\" \/> (c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-0869376fbbe578b58a8b97d9e44492e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#83;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#51;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"125\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector07\">\n<div class=\"bc-figcaption figcaption\">Vectors used in <a href=\"#fs-id1167132310249\" class=\"autogenerated-content\">(Figure)<\/a> and in the Check Your Understanding feature that follows.<\/div>\n<p><span id=\"fs-id1167132361887\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector07.jpg\" alt=\"Vector A has magnitude 10.0 and is at an angle alpha = 35 degrees counterclockwise from the horizontal. It points up and right. Vector B has magnitude 7.0 and is at an angle beta = -110 degrees clockwise from the horizontal. It points down and left. Vector C has magnitude 8.0 and is at an angle gamma = 30 degrees counterclockwise from the horizontal. It points up and right. Vector F has magnitude 20.0 and is at an angle phi = 110 degrees counterclockwise from the horizontal. It points up and left.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167132688124\"><span>Strategy<\/span><br \/>\nIn geometric construction, to find a vector means to find its magnitude and its direction angle with the horizontal direction. The strategy is to draw to scale the vectors that appear on the right-hand side of the equation and construct the resultant vector. Then, use a ruler and a protractor to read the magnitude of the resultant and the direction angle. For parts (a) and (b) we use the parallelogram rule. For (c) we use the tail-to-head method.<\/p>\n<p id=\"fs-id1167132688134\"><span>Solution<\/span><br \/>\nFor parts (a) and (b), we attach the origin of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> to the origin of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>, as shown in <a href=\"#CNX_UPhysics_02_01_vector08\" class=\"autogenerated-content\">(Figure)<\/a>, and construct a parallelogram. The shorter diagonal of this parallelogram is the sum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-aad8467ee9a8fc983f84246554c1a40a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"51\" style=\"vertical-align: -2px;\" \/>. The longer of the diagonals is the difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-3bdfba128e6d4730ad1ea6b78ad9af8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"51\" style=\"vertical-align: -2px;\" \/>. We use a ruler to measure the lengths of the diagonals, and a protractor to measure the angles with the horizontal. For the resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6bb3969d03d2d1cf2c9f16860552e6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, we obtain <em>R<\/em> = 5.8 cm and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-78aac11d7cb79d1a24e5f0032997104b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#82;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\" \/>. For the difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-526fa8444ce45845f3bb44c3dc6c4a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -2px;\" \/>, we obtain <em>D<\/em> = 16.2 cm and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ff5ac3be0fb5cd5052323295221bd7d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#68;&#125;&#61;&#52;&#57;&#46;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"76\" style=\"vertical-align: -3px;\" \/>, which are shown in <a href=\"#CNX_UPhysics_02_01_vector08\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector08\">\n<div class=\"bc-figcaption figcaption\">Using the parallelogram rule to solve (a) (finding the resultant, red) and (b) (finding the difference, blue).<\/div>\n<p><span id=\"fs-id1167132685555\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector08.jpg\" alt=\"Three diagrams of vectors A and B. Vectors A and B are shown placed tail to tail. Vector A points up and right and has magnitude 10.0. Vector B points down and left and has magnitude 7.0. The angle between vectors A and B is 145 degrees. In the second diagram, Vectors A and B are shown again along with the dashed lines completing the parallelogram. Vector R equaling the sum of vectors A and B is shown as the vector from the tails of A and B to the opposite vertex of the parallelogram. The magnitude of R is 5.8. In the third diagram, Vectors A and B are shown again along with the dashed lines completing the parallelogram. Vector D equaling the difference of vectors A and B is shown as the vector from the head of B to the head of A. The magnitude of D is 16.2, and the angle between D and the horizontal is 49.3 degrees. Vector R in the second diagram is much shorter than vector D in the third diagram.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167132451912\">For (c), we can start with vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-f3d4f59b6eaee94e9493ed5d8fdb69cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"41\" style=\"vertical-align: -2px;\" \/> and draw the remaining vectors tail-to-head as shown in <a href=\"#CNX_UPhysics_02_01_vector09\" class=\"autogenerated-content\">(Figure)<\/a>. In vector addition, the order in which we draw the vectors is unimportant, but drawing the vectors to scale is very important. Next, we draw vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b1b9e1f029dfcc753873f05a08c4a58c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> from the origin of the first vector to the end of the last vector and place the arrowhead at the end of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b1b9e1f029dfcc753873f05a08c4a58c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>. We use a ruler to measure the length of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b1b9e1f029dfcc753873f05a08c4a58c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>, and find that its magnitude is<\/p>\n<div><\/div>\n<p><em>S<\/em> = 36.9 cm. We use a protractor and find that its direction angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d4b24237799d010809abe3aea58b241a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#83;&#125;&#61;&#53;&#50;&#46;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -3px;\" \/>. This solution is shown in <a href=\"#CNX_UPhysics_02_01_vector09\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_02_01_vector09\">\n<div class=\"bc-figcaption figcaption\">Using the tail-to-head method to solve (c) (finding vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b1b9e1f029dfcc753873f05a08c4a58c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>, green).<\/div>\n<p><span id=\"fs-id1167132499445\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_vector09.jpg\" alt=\"Three vectors are shown in blue and placed head to tail: Vector minus 3 B points up and right and has magnitude 3 B = 21.0. Vector A starts at the head of B, points up and right, and has a magnitude of A=10.0. The angle between vector A and vector minus 3 B is 145 degrees. Vector C starts at the head of A and has magnitude C=8.0. Vector S is green and goes from the tail of minus 3 B to the head of C. Vector S equals vector A minus 3 vector B plus vector C, has a magnitude of S=36.9 and makes an angle of 52.9 degrees counterclockwise with the horizontal.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132487190\" class=\"check-understanding\">\n<div id=\"fs-id1167132487194\">\n<div id=\"fs-id1167132487196\">\n<p id=\"fs-id1167133787376\"><strong>Check Your Understanding<\/strong> Using the three displacement vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7247d9ea65980e227b547b3d37382865_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> in <a href=\"#CNX_UPhysics_02_01_vector07\" class=\"autogenerated-content\">(Figure)<\/a>, choose a convenient scale, and use a ruler and a protractor to find vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7a189778aae90b848c5e7c08d4b2edf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#71;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/> given by the vector equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-65fe1ff763fd82e26022e5fc46f362e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#71;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#50;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"126\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1167132326222\">\n<p id=\"fs-id1167132576679\"><em>G<\/em> = 28.2 cm, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-9658728394b29949af62e286076d8004_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#71;&#125;&#61;&#50;&#57;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133568457\" class=\"media-2\">\n<p id=\"fs-id1167133568461\">Observe the addition of vectors in a plane by visiting this <a href=\"https:\/\/openstaxcollege.org\/l\/21compveccalc\">vector calculator<\/a> and this <a href=\"https:\/\/openstaxcollege.org\/l\/21phetvecaddsim\">Phet simulation<\/a>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\" id=\"fs-id1167133772309\">\n<h3>Summary<\/h3>\n<ul id=\"fs-id1167132451699\">\n<li>A vector quantity is any quantity that has magnitude and direction, such as displacement or velocity. Vector quantities are represented by mathematical objects called vectors.<\/li>\n<li>Geometrically, vectors are represented by arrows, with the end marked by an arrowhead. The length of the vector is its magnitude, which is a positive scalar. On a plane, the direction of a vector is given by the angle the vector makes with a reference direction, often an angle with the horizontal. The direction angle of a vector is a scalar.<\/li>\n<li>Two vectors are equal if and only if they have the same magnitudes and directions. Parallel vectors have the same direction angles but may have different magnitudes. Antiparallel vectors have direction angles that differ by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-27bdda3969a0aa6f1afbe111d02e4073_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/>. Orthogonal vectors have direction angles that differ by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ad7fa5a1576b957af1a577f736d676e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>.<\/li>\n<li>When a vector is multiplied by a scalar, the result is another vector of a different length than the length of the original vector. Multiplication by a positive scalar does not change the original direction; only the magnitude is affected. Multiplication by a negative scalar reverses the original direction. The resulting vector is antiparallel to the original vector. Multiplication by a scalar is distributive. Vectors can be divided by nonzero scalars but cannot be divided by vectors.<\/li>\n<li>Two or more vectors can be added to form another vector. The vector sum is called the resultant vector. We can add vectors to vectors or scalars to scalars, but we cannot add scalars to vectors. Vector addition is commutative and associative.<\/li>\n<li>To construct a resultant vector of two vectors in a plane geometrically, we use the parallelogram rule. To construct a resultant vector of many vectors in a plane geometrically, we use the tail-to-head method.<\/li>\n<\/ul>\n<\/div>\n<div class=\"review-conceptual-questions\" id=\"fs-id1167132744637\">\n<h3>Conceptual Questions<\/h3>\n<div id=\"fs-id1167132744645\">\n<div id=\"fs-id1167132744647\">\n<p id=\"fs-id1167133708383\">A weather forecast states the temperature is predicted to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-417762b63b5c36389bf89a8026a7511d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: 0px;\" \/> the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1167133766192\">\n<p id=\"fs-id1167133766194\">scalar<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133578427\">\n<div id=\"fs-id1167133578429\">\n<p id=\"fs-id1167133578431\">Which of the following is a vector: a person\u2019s height, the altitude on Mt. Everest, the velocity of a fly, the age of Earth, the boiling point of water, the cost of a book, Earth\u2019s population, or the acceleration of gravity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132458328\">\n<div id=\"fs-id1167132458330\">\n<p id=\"fs-id1167132458332\">Give a specific example of a vector, stating its magnitude, units, and direction.<\/p>\n<\/div>\n<div id=\"fs-id1167132483960\">\n<p id=\"fs-id1167132483962\">answers may vary<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132483967\">\n<div id=\"fs-id1167132483969\">\n<p id=\"fs-id1167132230862\">What do vectors and scalars have in common? How do they differ?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132352997\">\n<div id=\"fs-id1167132352999\">\n<p id=\"fs-id1167132353001\">Suppose you add two vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8aa0cc7c8efa68a13f7384ca40b991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bc9a1897bfea81784605aed5fc42bf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -2px;\" \/>. What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?<\/p>\n<\/div>\n<div id=\"fs-id1167133640324\">\n<p id=\"fs-id1167133640326\">parallel, sum of magnitudes, antiparallel, zero<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133640332\">\n<div id=\"fs-id1167133640334\">\n<p id=\"fs-id1167133640336\">Is it possible to add a scalar quantity to a vector quantity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132315789\">\n<div id=\"fs-id1167132315791\">\n<p id=\"fs-id1167132415644\">Is it possible for two vectors of different magnitudes to add to zero? Is it possible for three vectors of different magnitudes to add to zero? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1167132415650\">\n<p id=\"fs-id1167132415652\">no, yes<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133635856\">\n<div id=\"fs-id1167133635858\">\n<p id=\"fs-id1167133635860\">Does the odometer in an automobile indicate a scalar or a vector quantity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167128860536\">\n<div id=\"fs-id1167128860538\">\n<p id=\"fs-id1167128860540\">When a 10,000-m runner competing on a 400-m track crosses the finish line, what is the runner\u2019s net displacement? Can this displacement be zero? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1167128860546\">\n<p id=\"fs-id1167132744080\">zero, yes<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132744085\">\n<div id=\"fs-id1167132744087\">\n<p id=\"fs-id1167132744089\">A vector has zero magnitude. Is it necessary to specify its direction? Explain.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132437699\">\n<div id=\"fs-id1167132437701\">\n<p id=\"fs-id1167132437703\">Can a magnitude of a vector be negative?<\/p>\n<\/div>\n<div id=\"fs-id1167132375593\">\n<p id=\"fs-id1167132375595\">no<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132375600\">\n<div id=\"fs-id1167132375603\">\n<p id=\"fs-id1167132341572\">Can the magnitude of a particle\u2019s displacement be greater that the distance traveled?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132318730\">\n<div id=\"fs-id1167132318732\">\n<p id=\"fs-id1167132318734\">If two vectors are equal, what can you say about their components? What can you say about their magnitudes? What can you say about their directions?<\/p>\n<\/div>\n<div id=\"fs-id1167132318740\">\n<p id=\"fs-id1167132318742\">equal, equal, the same<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132266747\">\n<div id=\"fs-id1167132266749\">\n<p id=\"fs-id1167132266751\">If three vectors sum up to zero, what geometric condition do they satisfy?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"review-problems\" id=\"fs-id1167132689349\">\n<h3>Problems<\/h3>\n<div id=\"fs-id1167132689356\">\n<div id=\"fs-id1167132255334\">\n<p id=\"fs-id1167132255336\">A scuba diver makes a slow descent into the depths of the ocean. His vertical position with respect to a boat on the surface changes several times. He makes the first stop 9.0 m from the boat but has a problem with equalizing the pressure, so he ascends 3.0 m and then continues descending for another 12.0 m to the second stop. From there, he ascends 4 m and then descends for 18.0 m, ascends again for 7 m and descends again for 24.0 m, where he makes a stop, waiting for his buddy. Assuming the positive direction up to the surface, express his net vertical displacement vector in terms of the unit vector. What is his distance to the boat?<\/p>\n<\/div>\n<div id=\"fs-id1167132255346\">\n<p id=\"fs-id1167133773306\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-1fa67c6b9b51de8e24bea56c10efdc21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#104;&#125;&#61;&#45;&#52;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"97\" style=\"vertical-align: -2px;\" \/>, 49 m<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132471146\">\n<div id=\"fs-id1167132471148\">\n<p id=\"fs-id1167132471150\">In a tug-of-war game on one campus, 15 students pull on a rope at both ends in an effort to displace the central knot to one side or the other. Two students pull with force 196 N each to the right, four students pull with force 98 N each to the left, five students pull with force 62 N each to the left, three students pull with force 150 N each to the right, and one student pulls with force 250 N to the left. Assuming the positive direction to the right, express the net pull on the knot in terms of the unit vector. How big is the net pull on the knot? In what direction?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132559167\">\n<div id=\"fs-id1167132559169\">\n<p id=\"fs-id1167132559171\">Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point and what is the compass direction of a line connecting your starting point to your final position? Use a graphical method.<\/p>\n<\/div>\n<div id=\"fs-id1167132559177\">\n<p id=\"fs-id1167132559180\">30.8 m, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-2fb53a777a883cf17c1e178015f92461_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#46;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> west of north<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132309641\">\n<div id=\"fs-id1167132296118\">\n<p id=\"fs-id1167132296120\">For the vectors given in the following figure, use a graphical method to find the following resultants: (a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-aad8467ee9a8fc983f84246554c1a40a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"51\" style=\"vertical-align: -2px;\" \/>, (b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-f3dee61ae09b51e54b3561e50b559fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#67;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -2px;\" \/>, (c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c177ab784dbc48b360b12781aed6729c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -2px;\" \/>, (d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-3bdfba128e6d4730ad1ea6b78ad9af8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"51\" style=\"vertical-align: -2px;\" \/>, (e) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-2499c11819e85893135e6574568a894e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#45;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -2px;\" \/>, (f) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-42e503a962ef48e7163ff6a66b039703_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#50;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -2px;\" \/>, (g); and (h) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-9910f5e7973887747401901d4d26b572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#45;&#52;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#68;&#125;&#43;&#50;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"108\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<p><span id=\"fs-id1167132294810\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_problems_img.jpg\" alt=\"The x y coordinate system is shown, with positive x to the right and positive y up. Vector A has magnitude 10.0 and makes an angle of 30 degrees above the positive x direction. Vector B has magnitude 5.0 and makes an angle of 53 degrees above the positive x direction. Vector C has magnitude 12.0 and makes an angle of 60 degrees below the positive x direction. Vector D has magnitude 20.0 and makes an angle of 37 degrees above the negative x direction. Vector F has magnitude 20.0 and makes an angle of 30 degrees below the negative x direction.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133740645\">\n<div id=\"fs-id1167133740647\">\n<p id=\"fs-id1167133740649\">A delivery man starts at the post office, drives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use a graphical method to find his net displacement vector.<\/p>\n<\/div>\n<div id=\"fs-id1167133740656\">\n<p id=\"fs-id1167133740658\">134 km, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-13b3a33ea4617093f551a50179e98256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132741855\">\n<div id=\"fs-id1167132741858\">\n<p id=\"fs-id1167132741860\">An adventurous dog strays from home, runs three blocks east, two blocks north, one block east, one block north, and two blocks west. Assuming that each block is about 100 m, how far from home and in what direction is the dog? Use a graphical method.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132674455\">\n<div id=\"fs-id1167132674457\">\n<p id=\"fs-id1167132674459\">In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day and he is blown along the following directions: 2.50 km and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e2ba96319cb2c1f14d5807ac0d28abb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/> north of west, then 4.70 km and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-1c0a419234aa489093a88305989d2c2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> south of east, then 1.30 km and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c403c9ab030a2c5041a41f8a959c0256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> south of west, then 5.10 km straight east, then 1.70 km and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-292f5694a964a52fb80a65d87c86196b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#48;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> east of north, then 7.20 km and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-2c9e78e9f1a086e5270e6e4a191ec1e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#53;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> south of west, and finally 2.80 km and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-cd00ec36c666dd2f5981083dd15041a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> north of east. Use a graphical method to find the castaway\u2019s final position relative to the island.<\/p>\n<\/div>\n<div id=\"fs-id1167133722463\">\n<p id=\"fs-id1167133722465\">7.34 km, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-623eb8b3d20394a3216e130f261298eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#51;&#46;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> south of east<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133766074\">\n<div id=\"fs-id1167133766076\">\n<p id=\"fs-id1167133766078\">A small plane flies 40.0 km in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-0932d0d7c22ee936ec17ee328e91c882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> north of east and then flies 30.0 km in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-0d33ef4cf8731157f3c2ac2d081030e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/> north of east. Use a graphical method to find the total distance the plane covers from the starting point and the direction of the path to the final position.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132546070\">\n<div id=\"fs-id1167132546072\">\n<p id=\"fs-id1167132546074\">A trapper walks a 5.0-km straight-line distance from his cabin to the lake, as shown in the following figure. Use a graphical method (the parallelogram rule) to determine the trapper\u2019s displacement directly to the east and displacement directly to the north that sum up to his resultant displacement vector. If the trapper walked only in directions east and north, zigzagging his way to the lake, how many kilometers would he have to walk to get to the lake?<\/p>\n<p><span id=\"fs-id1167132546077\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_02_01_problem3_img.jpg\" alt=\"North is up, east is to the right. A house and lake are shown. The x y coordiante system is also shown, with the origin near the house, the positive x direction to the right nad the positive y direction up. The vector from the house to the lake is shown as a straight red arrow, labeled as vector S, magnitude S=5.0 kilometers, and at an angle of 40 degrees above the posiitve x direction. Two meandering paths, path 1 and path 2, from the house to the lake are shown as dashed line.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1167132395714\">\n<p id=\"fs-id1167132395716\">3.8 km east, 3.2 km north, 7.0 km<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132395721\">\n<div id=\"fs-id1167132395723\">\n<p id=\"fs-id1167132395725\">A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c6eedb3526de357fd1a26da5d7ce355d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>. How wide is the river?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167132526027\">\n<div id=\"fs-id1167132526029\">\n<p id=\"fs-id1167132526031\">A pedestrian walks 6.0 km east and then 13.0 km north. Use a graphical method to find the pedestrian\u2019s resultant displacement and geographic direction.<\/p>\n<\/div>\n<div id=\"fs-id1167132526037\">\n<p id=\"fs-id1167133862600\">14.3 km, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4950e39bb1769ea30ab017d09b2018ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167133862614\">\n<div id=\"fs-id1167133862616\">\n<p id=\"fs-id1167133839798\">The magnitudes of two displacement vectors are <em>A<\/em> = 20 m and <em>B<\/em> = 6 m. What are the largest and the smallest values of the magnitude of the resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7d59b7aefcb2f931977455af3fdbd3e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#82;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#65;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#66;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1167132688517\">\n<dt>antiparallel vectors<\/dt>\n<dd id=\"fs-id1167132688522\">two vectors with directions that differ by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-27bdda3969a0aa6f1afbe111d02e4073_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579123\">\n<dt>associative<\/dt>\n<dd id=\"fs-id1167132579129\">terms can be grouped in any fashion<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579133\">\n<dt>commutative<\/dt>\n<dd id=\"fs-id1167132581795\">operations can be performed in any order<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132581799\">\n<dt>difference of two vectors<\/dt>\n<dd id=\"fs-id1167132581804\">vector sum of the first vector with the vector antiparallel to the second<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132581808\">\n<dt>displacement<\/dt>\n<dd id=\"fs-id1167133770991\">change in position<\/dd>\n<\/dl>\n<dl id=\"fs-id1167133770995\">\n<dt>distributive<\/dt>\n<dd id=\"fs-id1167133771000\">multiplication can be distributed over terms in summation<\/dd>\n<\/dl>\n<dl id=\"fs-id1167133771004\">\n<dt>magnitude<\/dt>\n<dd id=\"fs-id1167132504147\">length of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132504151\">\n<dt>orthogonal vectors<\/dt>\n<dd id=\"fs-id1167132504156\">two vectors with directions that differ by exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ad7fa5a1576b957af1a577f736d676e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>, synonymous with perpendicular vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132707485\">\n<dt>parallelogram rule<\/dt>\n<dd id=\"fs-id1167132707491\">geometric construction of the vector sum in a plane<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132520196\">\n<dt>parallel vectors<\/dt>\n<dd id=\"fs-id1167132520201\">two vectors with exactly the same direction angles<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132520205\">\n<dt>resultant vector<\/dt>\n<dd id=\"fs-id1167132520210\">vector sum of two (or more) vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167128844182\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id1167128844188\">a number, synonymous with a scalar quantity in physics<\/dd>\n<\/dl>\n<dl id=\"fs-id1167128844192\">\n<dt>scalar equation<\/dt>\n<dd id=\"fs-id1167128844197\">equation in which the left-hand and right-hand sides are numbers<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132586350\">\n<dt>scalar quantity<\/dt>\n<dd id=\"fs-id1167132586355\">quantity that can be specified completely by a single number with an appropriate physical unit<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132586360\">\n<dt>tail-to-head geometric construction<\/dt>\n<dd id=\"fs-id1167132586366\">geometric construction for drawing the resultant vector of many vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132550483\">\n<dt>unit vector<\/dt>\n<dd id=\"fs-id1167132550488\">vector of a unit magnitude that specifies direction; has no physical unit<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132550493\">\n<dt>vector<\/dt>\n<dd id=\"fs-id1167132550498\">mathematical object with magnitude and direction<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579716\">\n<dt>vector equation<\/dt>\n<dd id=\"fs-id1167132579722\">equation in which the left-hand and right-hand sides are vectors<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132579726\">\n<dt>vector quantity<\/dt>\n<dd id=\"fs-id1167132579731\">physical quantity described by a mathematical vector\u2014that is, by specifying both its magnitude and its direction; synonymous with a vector in physics<\/dd>\n<\/dl>\n<dl id=\"fs-id1167132445065\">\n<dt>vector sum<\/dt>\n<dd id=\"fs-id1167132445070\">resultant of the combination of two (or more) vectors<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-68","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":48,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/68","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions"}],"predecessor-version":[{"id":69,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions\/69"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/68\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/media?parent=68"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapter-type?post=68"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/contributor?post=68"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/license?post=68"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}