{"id":194,"date":"2017-10-27T16:29:06","date_gmt":"2017-10-27T16:29:06","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/vector-addition-and-subtraction-analytical-methods\/"},"modified":"2017-11-08T03:23:58","modified_gmt":"2017-11-08T03:23:58","slug":"vector-addition-and-subtraction-analytical-methods","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/vector-addition-and-subtraction-analytical-methods\/","title":{"raw":"Vector Addition and Subtraction: Analytical Methods","rendered":"Vector Addition and Subtraction: Analytical Methods"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Understand the rules of vector addition and subtraction using analytical methods.<\/li>\n<li>Apply analytical methods to determine vertical and horizontal component vectors.<\/li>\n<li>Apply analytical methods to determine the magnitude and direction of a resultant vector.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1165298787730\"><span data-type=\"term\" id=\"import-auto-id1165298770648\">Analytical methods<\/span> of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1472855\">\n<h1 data-type=\"title\">Resolving a Vector into Perpendicular Components<\/h1>\n<p id=\"import-auto-id1165298946816\">Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like [latex]\\mathbf{A}[\/latex] in <a href=\"#import-auto-id1165298677803\" class=\"autogenerated-content\">(Figure)<\/a>, we may wish to find which two perpendicular vectors, [latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{A}}_{y}[\/latex], add to produce it.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298677803\">\n<div class=\"bc-figcaption figcaption\">The vector [latex]\\mathbf{A}[\/latex], with its tail at the origin of an <em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>-coordinate system, is shown together with its <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-components, [latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{A}}_{y}[\/latex]. These vectors form a right triangle. The analytical relationships among these vectors are summarized below.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298881389\" data-alt=\"In the given figure a dotted vector A sub x is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A sub y at an angle theta from the x axis. On the graph a vector A, inclined at an angle theta with x axis is shown. Therefore vector A is the sum of the vectors A sub x and A sub y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"In the given figure a dotted vector A sub x is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A sub y at an angle theta from the x axis. On the graph a vector A, inclined at an angle theta with x axis is shown. Therefore vector A is the sum of the vectors A sub x and A sub y.\" height=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298681170\">[latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{A}}_{y}[\/latex]  are defined to be the components of <em data-effect=\"italics\">[latex]\\mathbf{A}[\/latex] along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes. The three vectors [latex]\\mathbf{A}[\/latex], [latex]{\\mathbf{A}}_{x}[\/latex], and [latex]{\\mathbf{A}}_{y}[\/latex] form a right triangle:<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-680\">[latex]{\\mathbf{A}}_{x}{\\text{&nbsp;+&nbsp;A}}_{y}\\text{&nbsp;=&nbsp;A}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1165298754511\">Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if [latex]{\\mathbf{\\text{A}}}_{x}=3 m[\/latex] east, <\/p>\n<p>[latex]{\\mathbf{\\text{A}}}_{y}=4 m[\/latex] north, and <\/p>\n<p>[latex]\\mathbf{\\text{A}}=5 m[\/latex] north-east, then it is true that the vectors [latex]{\\mathbf{A}}_{x}{\\text{&nbsp;+&nbsp;A}}_{y}\\text{&nbsp;=&nbsp;A}[\/latex]. However, it is <em data-effect=\"italics\"><em data-effect=\"italics\">not<\/em><\/em> true that the sum of the magnitudes of the vectors is also equal. That is,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-818\">[latex]\\begin{array}{}\\text{3 m}+\\text{4 m&nbsp;}\\ne \\text{&nbsp;5 m}\\\\ \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165298650894\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-505\">[latex]{A}_{x}+{A}_{y}\\ne A[\/latex]<\/div>\n<p id=\"import-auto-id1165298658167\">If the vector [latex]\\mathbf{A}[\/latex] is known, then its magnitude [latex]A[\/latex] (its length) and its angle  [latex]\\theta [\/latex] (its direction) are known. To find [latex]{A}_{x}[\/latex]<em data-effect=\"italics\"> and <em data-effect=\"italics\">[latex]{A}_{y}[\/latex], its <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components, we use the following relationships for a right triangle.<\/em><\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-377\">[latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex]<\/div>\n<p id=\"eip-25\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-69\">[latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\text{.}[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298704788\">\n<div class=\"bc-figcaption figcaption\">The magnitudes of the vector components [latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{A}}_{y}[\/latex] can be related to the resultant vector [latex]\\mathbf{A}[\/latex] and the angle  [latex]\\theta [\/latex] with trigonometric identities. Here we see that [latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and [latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298871520\" data-alt=\"]A dotted vector A sub x whose magnitude is equal to A cosine theta is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y whose magnitude is equal to A sine theta is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A-y at an angle theta from the x axis. Therefore vector A is the sum of the vectors A sub x and A sub y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"]A dotted vector A sub x whose magnitude is equal to A cosine theta is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y whose magnitude is equal to A sine theta is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A-y at an angle theta from the x axis. Therefore vector A is the sum of the vectors A sub x and A sub y.\" height=\"225\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298625515\">Suppose, for example, that [latex]\\mathbf{A}[\/latex] is the vector representing the total displacement of the person walking in a city considered in <a href=\"\/contents\/21d0e217-d50f-4901-af75-905e738eb4c4@4\">Kinematics in Two Dimensions: An Introduction<\/a> and <a href=\"\/contents\/4bd8bbec-bda2-412b-96f8-cc0b7ff5e794@7\">Vector Addition and Subtraction: Graphical Methods<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298544262\">\n<div class=\"bc-figcaption figcaption\">We can use the relationships [latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and [latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] to determine the magnitude of the horizontal and vertical component vectors in this example.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298994353\" data-alt=\"In the given figure a vector A of magnitude ten point three blocks is inclined at an angle twenty nine point one degrees to the positive x axis. The horizontal component A sub x of vector A is equal to A cosine theta which is equal to ten point three blocks multiplied to cosine twenty nine point one degrees which is equal to nine blocks east. Also the vertical component A sub y of vector A is equal to A sin theta is equal to ten point three blocks multiplied to sine twenty nine point one degrees,  which is equal to five point zero blocks north.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_03.jpg\" data-media-type=\"image\/jpg\" alt=\"In the given figure a vector A of magnitude ten point three blocks is inclined at an angle twenty nine point one degrees to the positive x axis. The horizontal component A sub x of vector A is equal to A cosine theta which is equal to ten point three blocks multiplied to cosine twenty nine point one degrees which is equal to nine blocks east. Also the vertical component A sub y of vector A is equal to A sin theta is equal to ten point three blocks multiplied to sine twenty nine point one degrees,  which is equal to five point zero blocks north.\" width=\"425\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165296377568\">Then [latex]A=10.3[\/latex] blocks and<br>\n[latex]\\theta =29.1\u00ba[\/latex]<br>\n, so that <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1646569\">[latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta =\\left(\\text{10.3 blocks}\\right)\\left(\\text{cos}\\phantom{\\rule{0.25em}{0ex}}29.1\u00ba\\right)=\\text{9.0 blocks}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id2495034\">[latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta =\\left(\\text{10.3 blocks}\\right)\\left(\\text{sin}\\phantom{\\rule{0.25em}{0ex}}29.1\u00ba\\right)=\\text{5.0 blocks}\\text{.}[\/latex]<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1344575\">\n<h1 data-type=\"title\">Calculating a Resultant Vector<\/h1>\n<p id=\"import-auto-id1165298995012\">If the perpendicular components [latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{A}}_{y}[\/latex] of a vector [latex]\\mathbf{A}[\/latex] are known, then [latex]\\mathbf{A}[\/latex] can also be found analytically. To find the magnitude [latex]A[\/latex] and direction  [latex]\\theta [\/latex] of a vector from its perpendicular components [latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{A}}_{y}[\/latex], we use the following relationships:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-109\">[latex]A=\\sqrt{{A}_{{x}^{2}}+{A}_{{y}^{2}}}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-750\">[latex]\\theta ={\\text{tan}}^{-1}\\left({A}_{y}\/{A}_{x}\\right)\\text{.}[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298723894\">\n<div class=\"bc-figcaption figcaption\">The magnitude and direction of the resultant vector  can be determined once the horizontal and vertical components [latex]{A}_{x}[\/latex] and [latex]{A}_{y}[\/latex]  have been determined.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296221052\" data-alt=\"Vector A is shown with its horizontal and vertical components A sub x and A sub y respectively. The magnitude of vector A is equal to the square root of A sub x squared plus A sub y squared. The angle theta of the vector A with the x axis is equal to inverse tangent of A sub y over A sub x\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_04a.jpg\" data-media-type=\"image\/png\" alt=\"Vector A is shown with its horizontal and vertical components A sub x and A sub y respectively. The magnitude of vector A is equal to the square root of A sub x squared plus A sub y squared. The angle theta of the vector A with the x axis is equal to inverse tangent of A sub y over A sub x\" width=\"145\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298666079\">Note that the equation<br>\n[latex]A=\\sqrt{{A}_{x}^{2}+{A}_{y}^{2}}[\/latex] is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if [latex]{A}_{x}[\/latex] and [latex]{A}_{y}[\/latex] are 9 and 5 blocks, respectively, then [latex]A=\\sqrt{{9}^{2}{\\text{+5}}^{2}}\\text{=10}\\text{.}3[\/latex] blocks, again consistent with the example of the person walking in a city. Finally, the direction is<br>\n[latex]\\theta ={\\text{tan}}^{\u20131}\\left(\\text{5\/9}\\right)=29.1\u00ba[\/latex]<\/p>\n<p>, as before.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1607685\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Determining Vectors and Vector Components with Analytical Methods<\/div>\n<p id=\"import-auto-id1165298788924\">Equations [latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and [latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] are used to find the perpendicular components of a vector\u2014that is, to go from [latex]A[\/latex] and  [latex]\\theta [\/latex] to [latex]{A}_{x}[\/latex] and [latex]{A}_{y}[\/latex]. Equations [latex]A=\\sqrt{{A}_{x}^{2}+{A}_{y}^{2}}[\/latex] and [latex]\\theta ={\\text{tan}}^{\\text{\u20131}}\\left({A}_{y}\/{A}_{x}\\right)[\/latex] are used to find a vector from its perpendicular components\u2014that is, to go from [latex]{A}_{x}[\/latex] and [latex]{A}_{y}[\/latex] to [latex]A[\/latex] and  [latex]\\theta [\/latex]. Both processes are crucial to analytical methods of vector addition and subtraction.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1461904\">\n<h1 data-type=\"title\">Adding Vectors Using Analytical Methods<\/h1>\n<p id=\"import-auto-id1165296570122\">To see how to add vectors using perpendicular components, consider <a href=\"#import-auto-id1165298839640\" class=\"autogenerated-content\">(Figure)<\/a>, in which the vectors [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] are added to produce the resultant [latex]\\mathbf{R}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298839640\">\n<div class=\"bc-figcaption figcaption\">Vectors [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] are two legs of a walk, and [latex]\\mathbf{R}[\/latex] is the resultant or total displacement. You can use analytical methods to determine the magnitude and direction of [latex]\\mathbf{R}[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298460444\" data-alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298462868\">If [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] represent two legs of a walk (two displacements), then [latex]\\mathbf{R}[\/latex] is the total displacement. The person taking the walk ends up at the tip of [latex]\\mathbf{R}.[\/latex] There are many ways to arrive at the same point. In particular, the person could have walked first in the <em data-effect=\"italics\">x<\/em>-direction and then in the <em data-effect=\"italics\">y<\/em>-direction. Those paths are the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-components of the resultant, [latex]{\\mathbf{R}}_{x}[\/latex] and [latex]{\\mathbf{R}}_{y}[\/latex]. If we know <\/p>\n<p>[latex]{\\mathbf{\\text{R}}}_{x}[\/latex] and [latex]{\\mathbf{R}}_{y}[\/latex], we can find <\/p>\n<p>[latex]R[\/latex] and  <\/p>\n<p>[latex]\\theta [\/latex] using the equations <\/p>\n<p>[latex]A=\\sqrt{{{A}_{x}}^{2}+{{A}_{y}}^{2}}[\/latex] and <\/p>\n<p>[latex]\\theta ={\\text{tan}}^{\u20131}\\left({A}_{y}\/{A}_{x}\\right)[\/latex]. When you use the analytical method of vector addition, you can determine the components or the magnitude and direction of a vector.<\/p>\n<p id=\"import-auto-id1165299001803\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 1.<\/strong><em data-effect=\"italics\"> Identify the x- and y-axes that will be used in the problem. Then, find the components of each vector to be added along the chosen perpendicular axes<\/em>.<\/em> Use the equations<br>\n[latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and <\/p>\n<p>[latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] to find the components. In <a href=\"#import-auto-id1165296674934\" class=\"autogenerated-content\">(Figure)<\/a>, these components are <\/p>\n<p>[latex]{A}_{x}[\/latex], <\/p>\n<p>[latex]{A}_{y}[\/latex], <\/p>\n<p>[latex]{B}_{x}[\/latex], and <\/p>\n<p>[latex]{B}_{y}[\/latex]. The angles that vectors [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] make with the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-axis are [latex]{\\theta }_{\\text{A}}[\/latex] and [latex]{\\theta }_{\\text{B}}[\/latex], respectively.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296674934\">\n<div class=\"bc-figcaption figcaption\">To add vectors [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex], first determine the horizontal and vertical components of each vector. These are the dotted vectors [latex]{\\mathbf{A}}_{x}[\/latex], [latex]{\\mathbf{A}}_{y}[\/latex], [latex]{\\mathbf{B}}_{x}[\/latex] and  [latex]{\\mathbf{\\text{B}}}_{y}[\/latex] shown in the image.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296674935\" data-alt=\"Two vectors A and B are shown. The tail of the vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The horizontal and vertical components of the vectors A and B are shown with the help of dotted lines. The vectors labeled as A sub x and A sub y are the components of vector A, and B sub x and B sub y as the components of vector B..\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of the vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The horizontal and vertical components of the vectors A and B are shown with the help of dotted lines. The vectors labeled as A sub x and A sub y are the components of vector A, and B sub x and B sub y as the components of vector B..\" width=\"325\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298948327\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 2.<\/strong><em data-effect=\"italics\"> Find the components of the resultant along each axis by adding the components of the individual vectors along that axis<\/em>.<\/em> That is, as shown in <a href=\"#import-auto-id1165298866862\" class=\"autogenerated-content\">(Figure)<\/a>,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-284\">[latex]{R}_{x}={A}_{x}+{B}_{x}[\/latex]<\/div>\n<p id=\"eip-342\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-92\">[latex]{R}_{y}={A}_{y}+{B}_{y}\\text{.}[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298866862\">\n<div class=\"bc-figcaption figcaption\">The magnitude of the vectors [latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{B}}_{x}[\/latex] add to give the magnitude [latex]{R}_{x}[\/latex] of the resultant vector in the horizontal direction. Similarly, the magnitudes of the vectors [latex]{\\mathbf{A}}_{y}[\/latex] and  [latex]{\\mathbf{B}}_{y}[\/latex] add to give the magnitude [latex]{R}_{y}[\/latex] of the resultant vector in the vertical direction.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298866864\" data-alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The vectors A and B are resolved into the horizontal and vertical components shown as dotted lines parallel to x axis and y axis respectively. The horizontal components of vector A and vector B are labeled as A sub x and B sub x and the horizontal component of the resultant R is labeled at R sub x and is equal to A sub x plus B sub x. The vertical components of vector A and vector B are labeled as A sub y and B sub y and the vertical components of the resultant R is labeled as R sub y is equal to A sub y plus B sub y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_07a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The vectors A and B are resolved into the horizontal and vertical components shown as dotted lines parallel to x axis and y axis respectively. The horizontal components of vector A and vector B are labeled as A sub x and B sub x and the horizontal component of the resultant R is labeled at R sub x and is equal to A sub x plus B sub x. The vertical components of vector A and vector B are labeled as A sub y and B sub y and the vertical components of the resultant R is labeled as R sub y is equal to A sub y plus B sub y.\" width=\"375\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165296251909\">Components along the same axis, say the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of [latex]\\mathbf{R}[\/latex] are known, its magnitude and direction can be found.<\/p>\n<p id=\"import-auto-id1165296245826\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 3.<\/strong><em data-effect=\"italics\"> To get the magnitude [latex]R[\/latex] of the resultant, use the Pythagorean theorem:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-960\">[latex]R=\\sqrt{{R}_{x}^{2}+{R}_{y}^{2}}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1165296334432\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 4.<\/strong><em data-effect=\"italics\"> To get the direction of the resultant:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-173\">[latex]\\theta ={\\text{tan}}^{-1}\\left({R}_{y}\/{R}_{x}\\right)\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1165298597164\">The following example illustrates this technique for adding vectors using perpendicular components.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1608746\">\n<div data-type=\"title\" class=\"title\">Adding Vectors Using Analytical Methods<\/div>\n<p id=\"import-auto-id1165298540622\">Add the vector [latex]\\mathbf{A}[\/latex] to the vector [latex]\\mathbf{B}[\/latex] shown in <a href=\"#import-auto-id1165296662297\" class=\"autogenerated-content\">(Figure)<\/a>, using perpendicular components along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes. The <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes are along the east\u2013west and north\u2013south directions, respectively. Vector [latex]\\mathbf{A}[\/latex] represents the first leg of a walk in which a person walks [latex]\\text{53}\\text{.}\\text{0 m}[\/latex] in a direction [latex]\\text{20}\\text{.}0\\text{\u00ba}[\/latex] north of east. Vector [latex]\\mathbf{B}[\/latex] represents the second leg, a displacement of [latex]\\text{34}\\text{.}\\text{0 m}[\/latex] in a direction [latex]\\text{63}\\text{.}0\\text{\u00ba}[\/latex] north of east.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296662297\">\n<div class=\"bc-figcaption figcaption\">Vector [latex]\\mathbf{A}[\/latex] has magnitude [latex]\\text{53}\\text{.}\\text{0 m}[\/latex] and direction [latex]\\text{20}\\text{.}0\u00ba[\/latex] north of the <em data-effect=\"italics\">x<\/em>-axis. Vector [latex]\\mathbf{B}[\/latex] has magnitude [latex]\\text{34}\\text{.}\\text{0 m}[\/latex] and direction [latex]\\text{63}\\text{.}0\\text{\u00ba}[\/latex] north of the <em data-effect=\"italics\">x<\/em>-axis. You can use analytical methods to determine the magnitude and direction of [latex]\\mathbf{R}[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298717013\" data-alt=\"Two vectors A and B are shown. The tail of the vector A is at origin. Both the vectors are in the first quadrant. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. From the head of the vector A another vector B of magnitude 34 units is drawn and is inclined at angle sixty three degrees with the horizontal. The resultant of two vectors is drawn from the tail of the vector A to the head of the vector B.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_08a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of the vector A is at origin. Both the vectors are in the first quadrant. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. From the head of the vector A another vector B of magnitude 34 units is drawn and is inclined at angle sixty three degrees with the horizontal. The resultant of two vectors is drawn from the tail of the vector A to the head of the vector B.\" width=\"325\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298560481\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1847310\">The components of [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes represent walking due east and due north to get to the same ending point. Once found, they are combined to produce the resultant.<\/p>\n<p id=\"import-auto-id1165298717036\"><strong>Solution<\/strong><\/p>\n<p id=\"fs-id1805726\">Following the method outlined above, we first find the components of <\/p>\n<p>[latex]\\mathbf{A}[\/latex] and <\/p>\n<p>[latex]\\mathbf{B}[\/latex] along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes. Note that <\/p>\n<p>[latex]A=53.0 m\\text{}[\/latex], <\/p>\n<p>[latex]{\\theta }_{\\text{A}}=20.0\u00ba[\/latex], <\/p>\n<p>[latex]B=34.0 m\\text{}[\/latex], and [latex]{\\theta }_{\\text{B}}=63.0\u00ba[\/latex].<\/p>\n<p> We find the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-components by using [latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex], which gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-253\">[latex]\\begin{array}{lll}{A}_{x}&amp; =&amp; A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{A}=\\left(\\text{53.}0 m\\text{}\\right)\\left(\\text{cos 20.0\u00ba}\\right)\\\\ &amp; =&amp; \\left(\\text{53.}0 m\\right)\\left(0\\text{.940}\\right)=\\text{49.}8 m\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165298793922\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-356\">[latex]\\begin{array}{lll}{B}_{x}&amp; =&amp; B\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{B}=\\left(\\text{34}\\text{.}0 m\\text{}\\right)\\left(\\text{cos 63.0\u00ba}\\right)\\\\ &amp; =&amp; \\text{}\\left(\\text{34}\\text{.}0 m\\text{}\\right)\\left(0\\text{.}\\text{454}\\right)=\\text{15}\\text{.}4 m\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165298886736\">Similarly, the <em data-effect=\"italics\">y<\/em>-components are found using [latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{A}[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-802\">[latex]\\begin{array}{lll}{A}_{y}&amp; =&amp; A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{A}=\\left(\\text{53}\\text{.}0 m\\text{}\\right)\\left(\\text{sin 20.0\u00ba}\\right)\\\\ &amp; =&amp; \\text{}\\left(\\text{53}\\text{.}0 m\\text{}\\right)\\left(0\\text{.}\\text{342}\\right)=\\text{18}\\text{.}1 m\\text{}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165298666019\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-837\">[latex]\\begin{array}{lll}{B}_{y}&amp; =&amp; B\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{B}=\\left(\\text{34}\\text{.}0 m\\text{}\\right)\\left(\\text{sin 63}\\text{.}0\u00ba\\right)\\\\ &amp; =&amp; \\text{}\\left(\\text{34}\\text{.}0 m\\text{}\\right)\\left(0\\text{.}\\text{891}\\right)=\\text{30}\\text{.}3 m\\text{}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165296543690\">The <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components of the resultant are thus<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-196\">[latex]{R}_{x}={A}_{x}+{B}_{x}=\\text{49}\\text{.}8 m\\text{}+\\text{15}\\text{.}4 m\\text{}=\\text{65}\\text{.}2 m\\text{}[\/latex]<\/div>\n<p id=\"import-auto-id1165298858089\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-325\">[latex]{R}_{y}={A}_{y}+{B}_{y}=\\text{18}\\text{.}1 m+\\text{30}\\text{.}3 m=\\text{48}\\text{.}4 m\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1165298673237\">Now we can find the magnitude of the resultant by using the Pythagorean theorem:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-941\">[latex]R=\\sqrt{{R}_{x}^{2}+{R}_{y}^{2}}=\\sqrt{\\left(\\text{65}\\text{.}2{\\right)}^{2}+\\left(\\text{48}\\text{.}4{\\right)}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{m}}[\/latex]<\/div>\n<p id=\"import-auto-id1165298868193\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]R=81.2 m.\\text{}[\/latex]<\/div>\n<p id=\"import-auto-id1165298940934\">Finally, we find the direction of the resultant:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-713\">[latex]\\theta ={\\text{tan}}^{-1}\\left({R}_{y}\/{R}_{x}\\right)\\text{=+}{\\text{tan}}^{-1}\\left(\\text{48}\\text{.}4\/\\text{65}\\text{.}2\\right)\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1165298853595\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-755\">[latex]\\theta ={\\text{tan}}^{-1}\\left(0\\text{.}\\text{742}\\right)=\\text{36}\\text{.}6\u00ba\\text{.}[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298804108\">\n<div class=\"bc-figcaption figcaption\">Using analytical methods, we see that the magnitude of [latex]\\mathbf{R}[\/latex] is [latex]\\text{81}\\text{.}\\text{2 m}[\/latex] and its direction is [latex]\\text{36}\\text{.}6\u00ba[\/latex] north of east.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298804109\" data-alt=\"The addition of two vectors A and B is shown. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. Vector B is of magnitude thirty four units and is inclined at angle sixty three degrees to the horizontal. The components of vector A are shown as dotted vectors A X is equal to forty nine point eight meter along x axis and A Y is equal to eighteen point one meter along Y axis. The components of vector B are also shown as dotted vectors B X is equal to fifteen point four meter and B Y is equal to thirty point three meter. The horizontal component of the resultant R X is equal to A X plus B X is equal to sixty five point two meter. The vertical component of the resultant R Y is equal to A Y plus B Y is equal to forty eight point four meter. The magnitude of the resultant of two vectors is eighty one point two meters. The direction of the resultant R is in thirty six point six degree from the vector A in anticlockwise direction.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_09a.jpg\" data-media-type=\"image\/jpg\" alt=\"The addition of two vectors A and B is shown. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. Vector B is of magnitude thirty four units and is inclined at angle sixty three degrees to the horizontal. The components of vector A are shown as dotted vectors A X is equal to forty nine point eight meter along x axis and A Y is equal to eighteen point one meter along Y axis. The components of vector B are also shown as dotted vectors B X is equal to fifteen point four meter and B Y is equal to thirty point three meter. The horizontal component of the resultant R X is equal to A X plus B X is equal to sixty five point two meter. The vertical component of the resultant R Y is equal to A Y plus B Y is equal to forty eight point four meter. The magnitude of the resultant of two vectors is eighty one point two meters. The direction of the resultant R is in thirty six point six degree from the vector A in anticlockwise direction.\" width=\"375\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298760698\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165298760703\">This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar\u2014it is just the addition of a negative vector.<\/p>\n<p id=\"import-auto-id1165298781739\">Subtraction of vectors is accomplished by the addition of a negative vector. That is, [latex]\\mathbf{A}-\\mathbf{B}\\equiv \\mathbf{A}+\\left(\\mathbf{\u2013B}\\right)[\/latex]. Thus, <em data-effect=\"italics\">the method for the subtraction of vectors using perpendicular components is identical to that for addition<\/em>. The components of <\/p>\n<p>[latex]\\mathbf{\\text{\u2013B}}[\/latex] are the negatives of the components of <\/p>\n<p>[latex]\\mathbf{B}[\/latex]. The <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components of the resultant [latex]\\mathbf{A}-\\text{B = R}[\/latex] are thus<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-772\">[latex]{R}_{x}={A}_{x}+\\left(\u2013{B}_{x}\\right)[\/latex]<\/div>\n<p id=\"import-auto-id1165298867012\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-532\">[latex]{R}_{y}={A}_{y}+\\left(\u2013{B}_{y}\\right)[\/latex]<\/div>\n<p id=\"import-auto-id1165296249538\">and the rest of the method outlined above is identical to that for addition. (See <a href=\"#import-auto-id1165298841604\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<\/div>\n<p id=\"import-auto-id1165296249543\">Analyzing vectors using perpendicular components is very useful in many areas of physics, because perpendicular quantities are often independent of one another. The next module, <a href=\"\/contents\/69062f44-56d2-4111-88ff-f599727c4ed1@12\">Projectile Motion<\/a>, is one of many in which using perpendicular components helps make the picture clear and simplifies the physics.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298841604\">\n<div class=\"bc-figcaption figcaption\">The subtraction of the two vectors shown in <a href=\"#import-auto-id1165298839640\" class=\"autogenerated-content\">(Figure)<\/a>. The components of [latex]\\mathbf{\\text{\u2013B}}[\/latex] are the negatives of the components of [latex]\\mathbf{B}[\/latex]. The method of subtraction is the same as that for addition.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298841605\" data-alt=\"In this figure, the subtraction of two vectors A and B is shown. A red colored vector A is inclined at an angle theta A to the positive of x axis. From the head of vector A a blue vector negative B is drawn. Vector B is in west of south direction. The resultant of the vector A and vector negative B is shown as a black vector R from the tail of vector A to the head of vector negative B. The resultant R is inclined to x axis at an angle theta below the x axis. The components of the vectors are also shown along the coordinate axes as dotted lines of their respective colors.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_10a.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure, the subtraction of two vectors A and B is shown. A red colored vector A is inclined at an angle theta A to the positive of x axis. From the head of vector A a blue vector negative B is drawn. Vector B is in west of south direction. The resultant of the vector A and vector negative B is shown as a black vector R from the tail of vector A to the head of vector negative B. The resultant R is inclined to x axis at an angle theta below the x axis. The components of the vectors are also shown along the coordinate axes as dotted lines of their respective colors.\" width=\"300\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-948\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Vector Addition<\/div>\n<p id=\"eip-id1169737855449\">Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id3192946\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/d88943c282e01cffec6d2979cf81b7476abac1ff\/vector-addition_en.jar\">Vector Addition<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_3.3\" data-alt=\"\"><a href=\"\/resources\/d88943c282e01cffec6d2979cf81b7476abac1ff\/vector-addition_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1628255\">\n<h1 data-type=\"title\">Summary<\/h1>\n<ul id=\"fs-id1628261\">\n<li id=\"import-auto-id1165296238488\">The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.<\/li>\n<li id=\"import-auto-id1165296238492\">The steps to add vectors [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] using the analytical method are as follows:\n<p id=\"import-auto-id1165298699809\">Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equations<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298699812\">[latex]\\begin{array}{lll}{A}_{x}&amp; =&amp; A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta \\\\ {B}_{x}&amp; =&amp; B\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165298717987\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298717989\">[latex]\\begin{array}{lll}{A}_{y}&amp; =&amp; A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\\\ {B}_{y}&amp; =&amp; B\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165296564556\">Step 2: Add the horizontal and vertical components of each vector to determine the components [latex]{R}_{x}[\/latex] and [latex]{R}_{y}[\/latex] of the resultant vector, [latex]\\mathbf{\\text{R}}[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298564746\">[latex]{R}_{x}={A}_{x}+{B}_{x}[\/latex]<\/div>\n<p id=\"import-auto-id1165298586325\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298586327\">[latex]{R}_{y}={A}_{y}+{B}_{y.}[\/latex]<\/div>\n<p id=\"import-auto-id1165296524905\">Step 3: Use the Pythagorean theorem to determine the magnitude, [latex]R[\/latex], of the resultant vector [latex]\\mathbf{\\text{R}}[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298936413\">[latex]R=\\sqrt{{R}_{x}^{2}+{R}_{y}^{2}}.[\/latex]<\/div>\n<p id=\"import-auto-id1165296242517\">Step 4: Use a trigonometric identity to determine the direction, [latex]\\theta [\/latex], of [latex]\\mathbf{\\text{R}}[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165296245925\">[latex]\\theta ={\\text{tan}}^{-1}\\left({R}_{y}\/{R}_{x}\\right).[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1611286\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611291\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611292\">\n<p id=\"import-auto-id1165296525953\">Suppose you add two vectors [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{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>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611340\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611341\">\n<p id=\"fs-id1611343\">Give an example of a nonzero vector that has a component of zero.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611347\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611348\">\n<p id=\"fs-id1611350\">Explain why a vector cannot have a component greater than its own magnitude.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611354\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611356\">\n<p id=\"fs-id1611358\">If the vectors [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] are perpendicular, what is the component of [latex]\\mathbf{A}[\/latex] along the direction of [latex]\\mathbf{B}[\/latex]? What is the component of [latex]\\mathbf{B}[\/latex] along the direction of [latex]\\mathbf{A}[\/latex]?\n      <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"eip-18\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611479\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611480\">\n<p id=\"import-auto-id1165298863759\">Find the following for path C in <a href=\"#import-auto-id1165298863773\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298863773\">\n<div class=\"bc-figcaption figcaption\">The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296384447\" data-alt=\"A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meter each. Four paths A B C and D are shown in different colors. The path c shown as blue extends to one block towards north, then five blocks towards east and then two blocks towards south then one block towards west and one block towards north and finally three blocks towards west. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish for path C.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_20a.jpg\" data-media-type=\"image\/wmf\" alt=\"A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meter each. Four paths A B C and D are shown in different colors. The path c shown as blue extends to one block towards north, then five blocks towards east and then two blocks towards south then one block towards west and one block towards north and finally three blocks towards west. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish for path C.\" width=\"300\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2052721\">\n<p id=\"import-auto-id1165298832178\">(a) 1.56 km<\/p>\n<p id=\"import-auto-id1165296578636\">(b) 120 m east<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1876099\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1876100\">\n<p id=\"import-auto-id1165296578640\">Find the following for path D in <a href=\"#import-auto-id1165298863773\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1751204\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1751205\">\n<p id=\"import-auto-id1165296578641\">Find the north and east components of the displacement from San Francisco to Sacramento shown in <a href=\"#import-auto-id1165298797444\" class=\"autogenerated-content\">(Figure)<\/a>.\n<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298797444\"><span data-type=\"media\" id=\"import-auto-id1165298797445\" data-alt=\"A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_19a.jpg\" data-media-type=\"image\/wmf\" alt=\"A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.\" width=\"250\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2406348\">\n<p id=\"eip-id2406350\">North-component 87.0 km, east-component 87.0 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-287\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-61\">\n<p id=\"import-auto-id1165298667360\">Solve the following problem using analytical techniques: 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? (If you represent the two legs of the walk as vector displacements [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex], as in <a href=\"#import-auto-id1165298935750\" class=\"autogenerated-content\">(Figure)<\/a>, then this problem asks you to find their sum [latex]\\mathbf{R}=\\mathbf{A}+\\mathbf{B}[\/latex].)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298935750\">\n<div class=\"bc-figcaption figcaption\">The two displacements [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] add to give a total displacement [latex]\\mathbf{R}[\/latex] having magnitude [latex]R[\/latex] and direction [latex]\\theta [\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298935751\" data-alt=\"In the given figure displacement of a person is shown. First movement of the person is shown as vector A from origin along negative x axis. He then turns to his right. His movement is now shown as a vertical vector in north direction. The displacement vector R is also shown. In the question you are asked to find the displacement of the person from the start to finish.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_21a.jpg\" data-media-type=\"image\/wmf\" alt=\"In the given figure displacement of a person is shown. First movement of the person is shown as vector A from origin along negative x axis. He then turns to his right. His movement is now shown as a vertical vector in north direction. The displacement vector R is also shown. In the question you are asked to find the displacement of the person from the start to finish.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298883930\">Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-430\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-400\">\n<p>Repeat <a href=\"#eip-287\" class=\"autogenerated-content\">(Figure)<\/a> using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result\u2014that is,<br>\n[latex]\\mathbf{\\text{B + A = A + B}}[\/latex].)  Discuss how taking another path to reach the same point might help to overcome an obstacle blocking you other path.\n  <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-24\">\n<p id=\"eip-157\">\n    30.8 m, 35.8 west of north\n  <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1862376\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1862377\">\n<p id=\"import-auto-id1165298883937\">You drive [latex]7\\text{.}\\text{50 km}[\/latex] in a straight line in a direction <\/p>\n<p>[latex]15\u00ba[\/latex]  east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1629683\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1629684\">\n<p id=\"import-auto-id1165298883936\">Do <a href=\"#eip-287\" class=\"autogenerated-content\">(Figure)<\/a> again using analytical techniques and change the second leg of the walk to [latex]\\text{25.0 m}[\/latex] straight south. (This is equivalent to subtracting [latex]\\mathbf{B}[\/latex] from [latex]\\mathbf{A}[\/latex] \u2014that is, finding<br>\n[latex]\\mathbf{\\text{R}}\\prime =\\mathbf{\\text{A \u2013 B}}[\/latex]) (b) Repeat again, but now you first walk [latex]\\text{25}\\text{.}\\text{0 m}[\/latex] north and then [latex]\\text{18}\\text{.}\\text{0 m}[\/latex] east. (This is equivalent to subtract [latex]\\mathbf{A}[\/latex] from [latex]\\mathbf{B}[\/latex] \u2014that is, to find [latex]\\mathbf{A}=\\mathbf{B}+\\mathbf{C}[\/latex]. Is that consistent with your result?)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1822911\">\n<p id=\"import-auto-id1165298883928\">(a) [latex]\\text{30}\\text{.}\\text{8 m}[\/latex], [latex]\\text{54}\\text{.}2\u00ba[\/latex] south of west<\/p>\n<p id=\"import-auto-id1165298676944\">(b) [latex]\\text{30}\\text{.}\\text{8 m}[\/latex], [latex]\\text{54}\\text{.}2\u00ba[\/latex] north of east<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1956316\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1956317\">\n<p id=\"import-auto-id1165298676945\">A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors [latex]\\mathbf{A}[\/latex] from [latex]\\mathbf{B}[\/latex] in <a href=\"#eip-id3165265\" class=\"autogenerated-content\">(Figure)<\/a>. She then correctly calculates the length and orientation of the third side [latex]\\text{C}[\/latex]. What is her result?<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id3165265\"><span data-type=\"media\" id=\"import-auto-id1165296607394\" data-alt=\"In the given figure the sides of a triangular piece of land are shown in vector form. West corner is at origin. A vector starts from the origin towards south east direction and makes an angle twenty-one degrees with the horizontal. Then from the head of this vector another vector B making an angle eleven degrees with the vertical is drawn upwards. Then another vector C from the head of the vector B to the tail of the initial vector is drawn. The length and orientation of side C is indicated as unknown, represented by a question mark.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_11a.jpg\" data-media-type=\"image\/wmf\" alt=\"In the given figure the sides of a triangular piece of land are shown in vector form. West corner is at origin. A vector starts from the origin towards south east direction and makes an angle twenty-one degrees with the horizontal. Then from the head of this vector another vector B making an angle eleven degrees with the vertical is drawn upwards. Then another vector C from the head of the vector B to the tail of the initial vector is drawn. The length and orientation of side C is indicated as unknown, represented by a question mark.\" width=\"250\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1874820\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1874821\">\n<p id=\"import-auto-id1165298676947\">You fly [latex]\\text{32}\\text{.}\\text{0 km}[\/latex] in a straight line in still air in the direction <\/p>\n<p>[latex]35.0\u00ba[\/latex] south of west. (a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.) (b) Find the distances you would have to fly first in a direction <\/p>\n<p>[latex]45.0\u00ba[\/latex] south of west and then in a direction <\/p>\n<p>[latex]45.0\u00ba[\/latex] west of north. These are the components of the displacement along a different set of axes\u2014one rotated <\/p>\n<p>[latex]45\u00ba[\/latex].<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1797961\">\n<p id=\"import-auto-id1165298676948\">18.4 km south, then 26.2 km west(b) 31.5 km at<br>\n[latex]45.0\u00ba[\/latex] south of west, then 5.56 km at<br>\n[latex]45.0\u00ba[\/latex] west of north<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-379\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-201\">\n<p id=\"eip-222\">A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as [latex]\\mathbf{A},[\/latex]<br>\n [latex]\\mathbf{B},[\/latex] and [latex]\\mathbf{C}[\/latex] in <a href=\"#import-auto-id1165298543237\" class=\"autogenerated-content\">(Figure)<\/a>, and then correctly calculates the length and orientation of the fourth side [latex]\\mathbf{D}[\/latex].<br>\n  What is his result?<\/p>\n<p id=\"eip-68\">\n<\/p><div class=\"bc-figure figure\" id=\"import-auto-id1165298543237\"><span data-type=\"media\" id=\"import-auto-id1165298608985\" data-alt=\"A quadrilateral with sides A, B, C, and D. A begins at the end of D and is 4 point seven zero kilometers  at an angle of 7 point 5 degrees south of west. B begins at the end of A and is 2 point four eight kilometers in a direction sixteen degrees west of north. C begins at the end of B and is 3 point zero 2 kilometers in a direction nineteen degrees north of west. D begins at the end of C and runs distance and direction that must be calculated\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_12a.jpg\" data-media-type=\"image\/wmf\" alt=\"A quadrilateral with sides A, B, C, and D. A begins at the end of D and is 4 point seven zero kilometers  at an angle of 7 point 5 degrees south of west. B begins at the end of A and is 2 point four eight kilometers in a direction sixteen degrees west of north. C begins at the end of B and is 3 point zero 2 kilometers in a direction nineteen degrees north of west. D begins at the end of C and runs distance and direction that must be calculated\" width=\"300\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1849688\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1849689\">\n<p id=\"import-auto-id11652986769482\">In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines: [latex]2\\text{.}\\text{50 km}[\/latex]<br>\n[latex]45.0\u00ba[\/latex]<\/p>\n<p> north of west; then <\/p>\n[latex]4\\text{.}\\text{70 km}[\/latex]\n<p>[latex]60.0\u00ba[\/latex] south of east; then <\/p>\n[latex]1.30\\phantom{\\rule{0.25em}{0ex}}\\text{km}[\/latex]\n<p>[latex]25.0\u00ba[\/latex] south of west; then <\/p>\n<p>[latex]5\\text{.}\\text{10 km}[\/latex] straight east; then <\/p>\n[latex]1.70\\phantom{\\rule{0.25em}{0ex}}\\text{km}[\/latex]\n<p>[latex]5.00\u00ba[\/latex] east of north; then <\/p>\n[latex]7\\text{.}\\text{20 km}[\/latex]\n<p>[latex]55.0\u00ba[\/latex] south of west; and finally <\/p>\n[latex]2\\text{.}\\text{80 km}[\/latex]\n<p>[latex]10.0\u00ba[\/latex] north of east. What is his final position relative to the island?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1930787\">\n<p id=\"import-auto-id11652986769483\">[latex]7\\text{.}\\text{34 km}[\/latex], [latex]\\text{63}\\text{.}5\u00ba[\/latex] south of east<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1955218\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1955219\">\n<p id=\"import-auto-id11652986769484\">Suppose a pilot flies [latex]\\text{40}\\text{.}\\text{0 km}[\/latex] in a direction [latex]\\text{60\u00ba}[\/latex] north of east and then flies [latex]\\text{30}\\text{.}\\text{0 km}[\/latex] in a direction [latex]\\text{15\u00ba}[\/latex] north of east as shown in <a href=\"#import-auto-id1165298708571\" class=\"autogenerated-content\">(Figure)<\/a>. Find her total distance [latex]R[\/latex] from the starting point and the direction [latex]\\theta [\/latex] of the straight-line path to the final position. Discuss qualitatively how this flight would be altered by a wind from the north and how the effect of the wind would depend on both wind speed and the speed of the plane relative to the air mass.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298708571\"><span data-type=\"media\" id=\"import-auto-id1165298708572\" data-alt=\"A triangle  defined by vectors A, B, and R. A begins at the origin and run forty kilometers in a direction sixty degrees north of east. B begins at the end of A and runs thirty kilometers in a direction fifteen degrees north of east. R is the resultant vector and runs from the origin (the beginning of A) to the end of B for a distance and in a direction theta that need to be calculated.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_13a.jpg\" data-media-type=\"image\/wmf\" alt=\"A triangle  defined by vectors A, B, and R. A begins at the origin and run forty kilometers in a direction sixty degrees north of east. B begins at the end of A and runs thirty kilometers in a direction fifteen degrees north of east. R is the resultant vector and runs from the origin (the beginning of A) to the end of B for a distance and in a direction theta that need to be calculated.\" width=\"275\"><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1165299001085\">\n<dt>analytical method<\/dt>\n<dd id=\"fs-id1544968\">the method of determining the magnitude and direction of a resultant vector using the Pythagorean theorem and trigonometric identities<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Understand the rules of vector addition and subtraction using analytical methods.<\/li>\n<li>Apply analytical methods to determine vertical and horizontal component vectors.<\/li>\n<li>Apply analytical methods to determine the magnitude and direction of a resultant vector.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1165298787730\"><span data-type=\"term\" id=\"import-auto-id1165298770648\">Analytical methods<\/span> of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1472855\">\n<h1 data-type=\"title\">Resolving a Vector into Perpendicular Components<\/h1>\n<p id=\"import-auto-id1165298946816\">Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> in <a href=\"#import-auto-id1165298677803\" class=\"autogenerated-content\">(Figure)<\/a>, we may wish to find which two perpendicular vectors, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>, add to produce it.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298677803\">\n<div class=\"bc-figcaption figcaption\">The vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/>, with its tail at the origin of an <em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>-coordinate system, is shown together with its <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-components, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>. These vectors form a right triangle. The analytical relationships among these vectors are summarized below.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298881389\" data-alt=\"In the given figure a dotted vector A sub x is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A sub y at an angle theta from the x axis. On the graph a vector A, inclined at an angle theta with x axis is shown. Therefore vector A is the sum of the vectors A sub x and A sub y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"In the given figure a dotted vector A sub x is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A sub y at an angle theta from the x axis. On the graph a vector A, inclined at an angle theta with x axis is shown. Therefore vector A is the sum of the vectors A sub x and A sub y.\" height=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298681170\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>  are defined to be the components of <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes. The three vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/> form a right triangle:<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-680\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c4bc385366ec089281c3708a307f8f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#43;&#32;&#65;&#125;&#125;&#95;&#123;&#121;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#61;&#32;&#65;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298754511\">Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-04539fb1264d2f9b344a986c6f922775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#125;&#95;&#123;&#120;&#125;&#61;&#51;&#32;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: -3px;\" \/> east, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2cb8064b65f76e4cb853bc1d887e3084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#125;&#95;&#123;&#121;&#125;&#61;&#52;&#32;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -6px;\" \/> north, and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-458426b69e7e349666869f3228072399_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#61;&#53;&#32;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/> north-east, then it is true that the vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4115af6a3948a0260578a53a44dbcb1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#43;&#32;&#65;&#125;&#125;&#95;&#123;&#121;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#61;&#32;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -6px;\" \/>. However, it is <em data-effect=\"italics\"><em data-effect=\"italics\">not<\/em><\/em> true that the sum of the magnitudes of the vectors is also equal. That is,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-818\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fea6b8bb8d41aa0981921eac5bef350_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;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#109;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#32;&#109;&#32;&#125;&#92;&#110;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#53;&#32;&#109;&#125;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1165298650894\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-505\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-00d060d4f819945a2a5e85650a48e9cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#43;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#92;&#110;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298658167\">If the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> is known, then its magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> (its length) and its angle  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> (its direction) are known. To find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d2106cbc0669aaf9e1d3bb5b96333fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/><em data-effect=\"italics\"> and <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e066fe81ed4f8facdef07ddf51c2529f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/>, its <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components, we use the following relationships for a right triangle.<\/em><\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ab5ac3fe943d7cfd86c538c9aeb2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"eip-25\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-69\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a27303e8eae3f438e5715d41f357360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298704788\">\n<div class=\"bc-figcaption figcaption\">The magnitudes of the vector components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/> can be related to the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and the angle  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> with trigonometric identities. Here we see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ab5ac3fe943d7cfd86c538c9aeb2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a81700950d2290414044809ca64a3d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -6px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298871520\" data-alt=\"]A dotted vector A sub x whose magnitude is equal to A cosine theta is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y whose magnitude is equal to A sine theta is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A-y at an angle theta from the x axis. Therefore vector A is the sum of the vectors A sub x and A sub y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"]A dotted vector A sub x whose magnitude is equal to A cosine theta is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y whose magnitude is equal to A sine theta is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A-y at an angle theta from the x axis. Therefore vector A is the sum of the vectors A sub x and A sub y.\" height=\"225\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298625515\">Suppose, for example, that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> is the vector representing the total displacement of the person walking in a city considered in <a href=\"\/contents\/21d0e217-d50f-4901-af75-905e738eb4c4@4\">Kinematics in Two Dimensions: An Introduction<\/a> and <a href=\"\/contents\/4bd8bbec-bda2-412b-96f8-cc0b7ff5e794@7\">Vector Addition and Subtraction: Graphical Methods<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298544262\">\n<div class=\"bc-figcaption figcaption\">We can use the relationships <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ab5ac3fe943d7cfd86c538c9aeb2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a81700950d2290414044809ca64a3d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -6px;\" \/> to determine the magnitude of the horizontal and vertical component vectors in this example.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298994353\" data-alt=\"In the given figure a vector A of magnitude ten point three blocks is inclined at an angle twenty nine point one degrees to the positive x axis. The horizontal component A sub x of vector A is equal to A cosine theta which is equal to ten point three blocks multiplied to cosine twenty nine point one degrees which is equal to nine blocks east. Also the vertical component A sub y of vector A is equal to A sin theta is equal to ten point three blocks multiplied to sine twenty nine point one degrees,  which is equal to five point zero blocks north.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_03.jpg\" data-media-type=\"image\/jpg\" alt=\"In the given figure a vector A of magnitude ten point three blocks is inclined at an angle twenty nine point one degrees to the positive x axis. The horizontal component A sub x of vector A is equal to A cosine theta which is equal to ten point three blocks multiplied to cosine twenty nine point one degrees which is equal to nine blocks east. Also the vertical component A sub y of vector A is equal to A sin theta is equal to ten point three blocks multiplied to sine twenty nine point one degrees,  which is equal to five point zero blocks north.\" width=\"425\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165296377568\">Then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39c95122b6bb76292c865a8a6a68a4f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#49;&#48;&#46;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"69\" style=\"vertical-align: -1px;\" \/> blocks and<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-32924b805a1585f760883700778da6b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#50;&#57;&#46;&#49;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"63\" style=\"vertical-align: -1px;\" \/><br \/>\n, so that <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1646569\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-236d9fad3520a04e7c7b70ca8182a5c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#46;&#51;&#32;&#98;&#108;&#111;&#99;&#107;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#57;&#46;&#49;&ordm;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#46;&#48;&#32;&#98;&#108;&#111;&#99;&#107;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"399\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id2495034\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8e2f68f76734bc02dd422d9ae3d84201_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#46;&#51;&#32;&#98;&#108;&#111;&#99;&#107;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#57;&#46;&#49;&ordm;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#32;&#98;&#108;&#111;&#99;&#107;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"399\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1344575\">\n<h1 data-type=\"title\">Calculating a Resultant Vector<\/h1>\n<p id=\"import-auto-id1165298995012\">If the perpendicular components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/> of a vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> are known, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> can also be found analytically. To find the magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> and direction  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> of a vector from its perpendicular components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>, we use the following relationships:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-109\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9d03bbde6ebdd5924439791de806e1b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#65;&#125;&#95;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#123;&#65;&#125;&#95;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"134\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08391bcaf64d8a7be1f0ea7a427a4bbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"151\" style=\"vertical-align: -6px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298723894\">\n<div class=\"bc-figcaption figcaption\">The magnitude and direction of the resultant vector  can be determined once the horizontal and vertical components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d2106cbc0669aaf9e1d3bb5b96333fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e066fe81ed4f8facdef07ddf51c2529f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/>  have been determined.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296221052\" data-alt=\"Vector A is shown with its horizontal and vertical components A sub x and A sub y respectively. The magnitude of vector A is equal to the square root of A sub x squared plus A sub y squared. The angle theta of the vector A with the x axis is equal to inverse tangent of A sub y over A sub x\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_04a.jpg\" data-media-type=\"image\/png\" alt=\"Vector A is shown with its horizontal and vertical components A sub x and A sub y respectively. The magnitude of vector A is equal to the square root of A sub x squared plus A sub y squared. The angle theta of the vector A with the x axis is equal to inverse tangent of A sub y over A sub x\" width=\"145\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298666079\">Note that the equation<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ec0f49b86554b63597314fbe27102ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"120\" style=\"vertical-align: -13px;\" \/> is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d2106cbc0669aaf9e1d3bb5b96333fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e066fe81ed4f8facdef07ddf51c2529f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/> are 9 and 5 blocks, respectively, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7a86985ce8b5cd4707bfe40965c62e70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#57;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#43;&#53;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -2px;\" \/> blocks, again consistent with the example of the person walking in a city. Finally, the direction is<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c72508b77b615fcdfa32e0224ec09995_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#47;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#57;&#46;&#49;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"174\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>, as before.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1607685\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Determining Vectors and Vector Components with Analytical Methods<\/div>\n<p id=\"import-auto-id1165298788924\">Equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ab5ac3fe943d7cfd86c538c9aeb2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a81700950d2290414044809ca64a3d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -6px;\" \/> are used to find the perpendicular components of a vector\u2014that is, to go from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d2106cbc0669aaf9e1d3bb5b96333fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e066fe81ed4f8facdef07ddf51c2529f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/>. Equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ec0f49b86554b63597314fbe27102ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"120\" style=\"vertical-align: -13px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c95653ca5d08d2d72350dc656ef3cb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"137\" style=\"vertical-align: -6px;\" \/> are used to find a vector from its perpendicular components\u2014that is, to go from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d2106cbc0669aaf9e1d3bb5b96333fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e066fe81ed4f8facdef07ddf51c2529f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>. Both processes are crucial to analytical methods of vector addition and subtraction.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1461904\">\n<h1 data-type=\"title\">Adding Vectors Using Analytical Methods<\/h1>\n<p id=\"import-auto-id1165296570122\">To see how to add vectors using perpendicular components, consider <a href=\"#import-auto-id1165298839640\" class=\"autogenerated-content\">(Figure)<\/a>, in which the vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> are added to produce the resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298839640\">\n<div class=\"bc-figcaption figcaption\">Vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> are two legs of a walk, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is the resultant or total displacement. You can use analytical methods to determine the magnitude and direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298460444\" data-alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298462868\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> represent two legs of a walk (two displacements), then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is the total displacement. The person taking the walk ends up at the tip of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-07995aaa9da84adecaed3d0b3b73e3ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> There are many ways to arrive at the same point. In particular, the person could have walked first in the <em data-effect=\"italics\">x<\/em>-direction and then in the <em data-effect=\"italics\">y<\/em>-direction. Those paths are the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-components of the resultant, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-765a3c84a595420d01f761fc07bf5089_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c01a80e437c663b542f79f1ccc4742d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>. If we know <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b50af646e491257ff52d4e796171f56f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c01a80e437c663b542f79f1ccc4742d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>, we can find <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and  <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> using the equations <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dab7c0249ed2c4f1a35e2df40e66f6e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"136\" style=\"vertical-align: -11px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a61d47ea1ca5c6652dfb803e28adcaa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"143\" style=\"vertical-align: -6px;\" \/>. When you use the analytical method of vector addition, you can determine the components or the magnitude and direction of a vector.<\/p>\n<p id=\"import-auto-id1165299001803\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 1.<\/strong><em data-effect=\"italics\"> Identify the x- and y-axes that will be used in the problem. Then, find the components of each vector to be added along the chosen perpendicular axes<\/em>.<\/em> Use the equations<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ab5ac3fe943d7cfd86c538c9aeb2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a81700950d2290414044809ca64a3d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -6px;\" \/> to find the components. In <a href=\"#import-auto-id1165296674934\" class=\"autogenerated-content\">(Figure)<\/a>, these components are <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d2106cbc0669aaf9e1d3bb5b96333fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/>, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e066fe81ed4f8facdef07ddf51c2529f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/>, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1f2b6d31d2a5a79393a8cbcc791c2d81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#66;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/>, and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-52162d795208c8f556ae4c0b08a3a593_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#66;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/>. The angles that vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> make with the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-axis are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dcc4e2031723b02223e0b4f53c9dd112_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b83aab8bf071b98462e3332d79c0cd13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\" \/>, respectively.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296674934\">\n<div class=\"bc-figcaption figcaption\">To add vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, first determine the horizontal and vertical components of each vector. These are the dotted vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e17b9d3439d9d70842bbb55a15ad8e40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce16cc6aaee77a5bb8a044d661c075f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/> shown in the image.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296674935\" data-alt=\"Two vectors A and B are shown. The tail of the vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The horizontal and vertical components of the vectors A and B are shown with the help of dotted lines. The vectors labeled as A sub x and A sub y are the components of vector A, and B sub x and B sub y as the components of vector B..\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of the vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The horizontal and vertical components of the vectors A and B are shown with the help of dotted lines. The vectors labeled as A sub x and A sub y are the components of vector A, and B sub x and B sub y as the components of vector B..\" width=\"325\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298948327\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 2.<\/strong><em data-effect=\"italics\"> Find the components of the resultant along each axis by adding the components of the individual vectors along that axis<\/em>.<\/em> That is, as shown in <a href=\"#import-auto-id1165298866862\" class=\"autogenerated-content\">(Figure)<\/a>,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-284\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6701373235412c95c6c9d2547cc05945_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#43;&#123;&#66;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"eip-342\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-92\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9d30871d8f0ec2ad5beab5449cc24d63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#43;&#123;&#66;&#125;&#95;&#123;&#121;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298866862\">\n<div class=\"bc-figcaption figcaption\">The magnitude of the vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e17b9d3439d9d70842bbb55a15ad8e40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> add to give the magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d145dc4cb226f69d6e9bfe4abf13775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> of the resultant vector in the horizontal direction. Similarly, the magnitudes of the vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/> and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f62c26fc4d70e8ca3ff40b2f65256e1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/> add to give the magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03308bd57741deda7ad9413c7b235f03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/> of the resultant vector in the vertical direction.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298866864\" data-alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The vectors A and B are resolved into the horizontal and vertical components shown as dotted lines parallel to x axis and y axis respectively. The horizontal components of vector A and vector B are labeled as A sub x and B sub x and the horizontal component of the resultant R is labeled at R sub x and is equal to A sub x plus B sub x. The vertical components of vector A and vector B are labeled as A sub y and B sub y and the vertical components of the resultant R is labeled as R sub y is equal to A sub y plus B sub y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_07a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The vectors A and B are resolved into the horizontal and vertical components shown as dotted lines parallel to x axis and y axis respectively. The horizontal components of vector A and vector B are labeled as A sub x and B sub x and the horizontal component of the resultant R is labeled at R sub x and is equal to A sub x plus B sub x. The vertical components of vector A and vector B are labeled as A sub y and B sub y and the vertical components of the resultant R is labeled as R sub y is equal to A sub y plus B sub y.\" width=\"375\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165296251909\">Components along the same axis, say the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> are known, its magnitude and direction can be found.<\/p>\n<p id=\"import-auto-id1165296245826\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 3.<\/strong><em data-effect=\"italics\"> To get the magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> of the resultant, use the Pythagorean theorem:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-960\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76388e8026ffcf84563363b655011707_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"124\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"import-auto-id1165296334432\"><em data-effect=\"italics\"><strong data-effect=\"bold\">Step 4.<\/strong><em data-effect=\"italics\"> To get the direction of the resultant:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-173\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e10eb1f5397ef458b5933755261c0d33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298597164\">The following example illustrates this technique for adding vectors using perpendicular components.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1608746\">\n<div data-type=\"title\" class=\"title\">Adding Vectors Using Analytical Methods<\/div>\n<p id=\"import-auto-id1165298540622\">Add the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> to the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> shown in <a href=\"#import-auto-id1165296662297\" class=\"autogenerated-content\">(Figure)<\/a>, using perpendicular components along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes. The <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes are along the east\u2013west and north\u2013south directions, respectively. Vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> represents the first leg of a walk in which a person walks <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd36a45b13382fc005a0f752a8cfc828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/> in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ff48d8dfc1cbe5fd2cfa4e77982c185c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> north of east. Vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> represents the second leg, a displacement of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4218678b6135ee69c588ed82fca99faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d3f58f884c36240b283494b52d58dbbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> north of east.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296662297\">\n<div class=\"bc-figcaption figcaption\">Vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> has magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd36a45b13382fc005a0f752a8cfc828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/> and direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a966c0085719b08e1d1ef175a4b51ca3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> north of the <em data-effect=\"italics\">x<\/em>-axis. Vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> has magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4218678b6135ee69c588ed82fca99faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> and direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d3f58f884c36240b283494b52d58dbbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> north of the <em data-effect=\"italics\">x<\/em>-axis. You can use analytical methods to determine the magnitude and direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298717013\" data-alt=\"Two vectors A and B are shown. The tail of the vector A is at origin. Both the vectors are in the first quadrant. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. From the head of the vector A another vector B of magnitude 34 units is drawn and is inclined at angle sixty three degrees with the horizontal. The resultant of two vectors is drawn from the tail of the vector A to the head of the vector B.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_08a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors A and B are shown. The tail of the vector A is at origin. Both the vectors are in the first quadrant. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. From the head of the vector A another vector B of magnitude 34 units is drawn and is inclined at angle sixty three degrees with the horizontal. The resultant of two vectors is drawn from the tail of the vector A to the head of the vector B.\" width=\"325\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298560481\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1847310\">The components of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes represent walking due east and due north to get to the same ending point. Once found, they are combined to produce the resultant.<\/p>\n<p id=\"import-auto-id1165298717036\"><strong>Solution<\/strong><\/p>\n<p id=\"fs-id1805726\">Following the method outlined above, we first find the components of <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes. Note that <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1820af7878aaa21bb02d1204d674362d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#53;&#51;&#46;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"84\" style=\"vertical-align: 0px;\" \/>, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5a1c7c597b5a388e53f94e096d4699b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#61;&#50;&#48;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"75\" style=\"vertical-align: -4px;\" \/>, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-755042d2db615965f50df569df3d8c23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#61;&#51;&#52;&#46;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"85\" style=\"vertical-align: -1px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d321973127a08f31ab0cffc96f67d2fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#61;&#54;&#51;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"75\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p> We find the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-components by using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ab5ac3fe943d7cfd86c538c9aeb2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\" \/>, which gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-253\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d86ae4baf28c5495f1001106b88d3660_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;&#108;&#108;&#108;&#125;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#38;&#32;&#61;&#38;&#32;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#65;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#32;&#50;&#48;&#46;&#48;&ordm;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#46;&#125;&#48;&#32;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#57;&#52;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#46;&#125;&#56;&#32;&#109;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"295\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id1165298793922\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-356\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b6eda1fa76dbb57e657def71b52bb3e_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;&#108;&#108;&#108;&#125;&#123;&#66;&#125;&#95;&#123;&#120;&#125;&#38;&#32;&#61;&#38;&#32;&#66;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#66;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#32;&#54;&#51;&#46;&#48;&ordm;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"297\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id1165298886736\">Similarly, the <em data-effect=\"italics\">y<\/em>-components are found using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a3c93adf361cb1172c2216bc4d331ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -6px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-802\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-739f2a9824c804b054c6ffda3270ddcc_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;&#108;&#108;&#108;&#125;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#38;&#32;&#61;&#38;&#32;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#65;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#32;&#50;&#48;&#46;&#48;&ordm;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"291\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id1165298666019\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-837\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7072cb320d625fb3ad2d5e1138eaff35_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;&#108;&#108;&#108;&#125;&#123;&#66;&#125;&#95;&#123;&#121;&#125;&#38;&#32;&#61;&#38;&#32;&#66;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#66;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#32;&#54;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#57;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"293\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id1165296543690\">The <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components of the resultant are thus<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-196\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b1ce8be61751d9c200033c73a970b150_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#43;&#123;&#66;&#125;&#95;&#123;&#120;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#56;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#50;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"321\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1165298858089\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e15bde91811cc538a6d3a36bdcd0812_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#43;&#123;&#66;&#125;&#95;&#123;&#121;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&#32;&#109;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&#32;&#109;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"324\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298673237\">Now we can find the magnitude of the resultant by using the Pythagorean theorem:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-941\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-008ac143cfe63ae442d54464fb8ea427_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#50;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"312\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"import-auto-id1165298868193\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4f89d2f29de43a2b305e05015e353528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#56;&#49;&#46;&#50;&#32;&#109;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"89\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"import-auto-id1165298940934\">Finally, we find the direction of the resultant:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-713\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a0cd38448668e74dac0fa7698231c287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#43;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"316\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298853595\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-755\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-35ec749c5174a42d53e6fcdf1b7946fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#52;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#54;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"193\" style=\"vertical-align: -4px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298804108\">\n<div class=\"bc-figcaption figcaption\">Using analytical methods, we see that the magnitude of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ca2fc8656d70b823218abd565252391e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> and its direction is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-36d30b152186c451ad38f3e8034052b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#54;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> north of east.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298804109\" data-alt=\"The addition of two vectors A and B is shown. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. Vector B is of magnitude thirty four units and is inclined at angle sixty three degrees to the horizontal. The components of vector A are shown as dotted vectors A X is equal to forty nine point eight meter along x axis and A Y is equal to eighteen point one meter along Y axis. The components of vector B are also shown as dotted vectors B X is equal to fifteen point four meter and B Y is equal to thirty point three meter. The horizontal component of the resultant R X is equal to A X plus B X is equal to sixty five point two meter. The vertical component of the resultant R Y is equal to A Y plus B Y is equal to forty eight point four meter. The magnitude of the resultant of two vectors is eighty one point two meters. The direction of the resultant R is in thirty six point six degree from the vector A in anticlockwise direction.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_09a.jpg\" data-media-type=\"image\/jpg\" alt=\"The addition of two vectors A and B is shown. Vector A is of magnitude fifty three units and is inclined at an angle of twenty degrees to the horizontal. Vector B is of magnitude thirty four units and is inclined at angle sixty three degrees to the horizontal. The components of vector A are shown as dotted vectors A X is equal to forty nine point eight meter along x axis and A Y is equal to eighteen point one meter along Y axis. The components of vector B are also shown as dotted vectors B X is equal to fifteen point four meter and B Y is equal to thirty point three meter. The horizontal component of the resultant R X is equal to A X plus B X is equal to sixty five point two meter. The vertical component of the resultant R Y is equal to A Y plus B Y is equal to forty eight point four meter. The magnitude of the resultant of two vectors is eighty one point two meters. The direction of the resultant R is in thirty six point six degree from the vector A in anticlockwise direction.\" width=\"375\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298760698\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165298760703\">This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar\u2014it is just the addition of a negative vector.<\/p>\n<p id=\"import-auto-id1165298781739\">Subtraction of vectors is accomplished by the addition of a negative vector. That is, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-327ba71cef02568d29508b61e48204ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#92;&#101;&#113;&#117;&#105;&#118;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#45;&#66;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\" \/>. Thus, <em data-effect=\"italics\">the method for the subtraction of vectors using perpendicular components is identical to that for addition<\/em>. The components of <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0b8ffca5f402dba3bec8b1bdd435add_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> are the negatives of the components of <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components of the resultant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d772822ce355b3a8371490331bfb298d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#32;&#61;&#32;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"88\" style=\"vertical-align: -1px;\" \/> are thus<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-772\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-659296b50e62c452016fd19fd7692f3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#123;&#66;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1165298867012\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e1fbe2667ac5c053008c9fe02ad9e66f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#123;&#66;&#125;&#95;&#123;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"136\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165296249538\">and the rest of the method outlined above is identical to that for addition. (See <a href=\"#import-auto-id1165298841604\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<\/div>\n<p id=\"import-auto-id1165296249543\">Analyzing vectors using perpendicular components is very useful in many areas of physics, because perpendicular quantities are often independent of one another. The next module, <a href=\"\/contents\/69062f44-56d2-4111-88ff-f599727c4ed1@12\">Projectile Motion<\/a>, is one of many in which using perpendicular components helps make the picture clear and simplifies the physics.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298841604\">\n<div class=\"bc-figcaption figcaption\">The subtraction of the two vectors shown in <a href=\"#import-auto-id1165298839640\" class=\"autogenerated-content\">(Figure)<\/a>. The components of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0b8ffca5f402dba3bec8b1bdd435add_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> are the negatives of the components of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The method of subtraction is the same as that for addition.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298841605\" data-alt=\"In this figure, the subtraction of two vectors A and B is shown. A red colored vector A is inclined at an angle theta A to the positive of x axis. From the head of vector A a blue vector negative B is drawn. Vector B is in west of south direction. The resultant of the vector A and vector negative B is shown as a black vector R from the tail of vector A to the head of vector negative B. The resultant R is inclined to x axis at an angle theta below the x axis. The components of the vectors are also shown along the coordinate axes as dotted lines of their respective colors.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_10a.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure, the subtraction of two vectors A and B is shown. A red colored vector A is inclined at an angle theta A to the positive of x axis. From the head of vector A a blue vector negative B is drawn. Vector B is in west of south direction. The resultant of the vector A and vector negative B is shown as a black vector R from the tail of vector A to the head of vector negative B. The resultant R is inclined to x axis at an angle theta below the x axis. The components of the vectors are also shown along the coordinate axes as dotted lines of their respective colors.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-948\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Vector Addition<\/div>\n<p id=\"eip-id1169737855449\">Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id3192946\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/d88943c282e01cffec6d2979cf81b7476abac1ff\/vector-addition_en.jar\">Vector Addition<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_3.3\" data-alt=\"\"><a href=\"\/resources\/d88943c282e01cffec6d2979cf81b7476abac1ff\/vector-addition_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1628255\">\n<h1 data-type=\"title\">Summary<\/h1>\n<ul id=\"fs-id1628261\">\n<li id=\"import-auto-id1165296238488\">The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.<\/li>\n<li id=\"import-auto-id1165296238492\">The steps to add vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> using the analytical method are as follows:\n<p id=\"import-auto-id1165298699809\">Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equations<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298699812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f489a98a85ead68892f71bedd6773b06_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;&#108;&#108;&#108;&#125;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#38;&#32;&#61;&#38;&#32;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#92;&#32;&#123;&#66;&#125;&#95;&#123;&#120;&#125;&#38;&#32;&#61;&#38;&#32;&#66;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"124\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"import-auto-id1165298717987\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298717989\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1374cb912bc5ff5934325cef33b52cab_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;&#108;&#108;&#108;&#125;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#38;&#32;&#61;&#38;&#32;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#92;&#32;&#123;&#66;&#125;&#95;&#123;&#121;&#125;&#38;&#32;&#61;&#38;&#32;&#66;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"126\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"import-auto-id1165296564556\">Step 2: Add the horizontal and vertical components of each vector to determine the components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d145dc4cb226f69d6e9bfe4abf13775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03308bd57741deda7ad9413c7b235f03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -6px;\" \/> of the resultant vector, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-779de0898c3e2d61620e60760225bc65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298564746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6701373235412c95c6c9d2547cc05945_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#43;&#123;&#66;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1165298586325\">and<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298586327\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce048ede9db84e5028dca2362432d3df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#61;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#43;&#123;&#66;&#125;&#95;&#123;&#121;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165296524905\">Step 3: Use the Pythagorean theorem to determine the magnitude, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, of the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-779de0898c3e2d61620e60760225bc65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165298936413\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2f65a33d9dc09f94325e5f52b5911f69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"124\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"import-auto-id1165296242517\">Step 4: Use a trigonometric identity to determine the direction, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-779de0898c3e2d61620e60760225bc65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1165296245925\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-805c8a9c55cec7e423944bcaab967cbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#82;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1611286\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611291\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611292\">\n<p id=\"import-auto-id1165296525953\">Suppose you add two vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. 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>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611340\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611341\">\n<p id=\"fs-id1611343\">Give an example of a nonzero vector that has a component of zero.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611347\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611348\">\n<p id=\"fs-id1611350\">Explain why a vector cannot have a component greater than its own magnitude.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611354\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611356\">\n<p id=\"fs-id1611358\">If the vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> are perpendicular, what is the component of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> along the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>? What is the component of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> along the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/>?\n      <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"eip-18\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1611479\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1611480\">\n<p id=\"import-auto-id1165298863759\">Find the following for path C in <a href=\"#import-auto-id1165298863773\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298863773\">\n<div class=\"bc-figcaption figcaption\">The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296384447\" data-alt=\"A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meter each. Four paths A B C and D are shown in different colors. The path c shown as blue extends to one block towards north, then five blocks towards east and then two blocks towards south then one block towards west and one block towards north and finally three blocks towards west. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish for path C.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_20a.jpg\" data-media-type=\"image\/wmf\" alt=\"A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meter each. Four paths A B C and D are shown in different colors. The path c shown as blue extends to one block towards north, then five blocks towards east and then two blocks towards south then one block towards west and one block towards north and finally three blocks towards west. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish for path C.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2052721\">\n<p id=\"import-auto-id1165298832178\">(a) 1.56 km<\/p>\n<p id=\"import-auto-id1165296578636\">(b) 120 m east<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1876099\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1876100\">\n<p id=\"import-auto-id1165296578640\">Find the following for path D in <a href=\"#import-auto-id1165298863773\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1751204\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1751205\">\n<p id=\"import-auto-id1165296578641\">Find the north and east components of the displacement from San Francisco to Sacramento shown in <a href=\"#import-auto-id1165298797444\" class=\"autogenerated-content\">(Figure)<\/a>.\n<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298797444\"><span data-type=\"media\" id=\"import-auto-id1165298797445\" data-alt=\"A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_19a.jpg\" data-media-type=\"image\/wmf\" alt=\"A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.\" width=\"250\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2406348\">\n<p id=\"eip-id2406350\">North-component 87.0 km, east-component 87.0 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-287\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-61\">\n<p id=\"import-auto-id1165298667360\">Solve the following problem using analytical techniques: 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? (If you represent the two legs of the walk as vector displacements <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, as in <a href=\"#import-auto-id1165298935750\" class=\"autogenerated-content\">(Figure)<\/a>, then this problem asks you to find their sum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c4c5b86daddc64261dc3bd4a01901b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"90\" style=\"vertical-align: -2px;\" \/>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298935750\">\n<div class=\"bc-figcaption figcaption\">The two displacements <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> add to give a total displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-848d0d650a6beb7d86c8eebd735712be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> having magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298935751\" data-alt=\"In the given figure displacement of a person is shown. First movement of the person is shown as vector A from origin along negative x axis. He then turns to his right. His movement is now shown as a vertical vector in north direction. The displacement vector R is also shown. In the question you are asked to find the displacement of the person from the start to finish.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_21a.jpg\" data-media-type=\"image\/wmf\" alt=\"In the given figure displacement of a person is shown. First movement of the person is shown as vector A from origin along negative x axis. He then turns to his right. His movement is now shown as a vertical vector in north direction. The displacement vector R is also shown. In the question you are asked to find the displacement of the person from the start to finish.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298883930\">Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-430\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-400\">\n<p>Repeat <a href=\"#eip-287\" class=\"autogenerated-content\">(Figure)<\/a> using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result\u2014that is,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-16c8eb3d7367f0a72970337b614eab5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#32;&#43;&#32;&#65;&#32;&#61;&#32;&#65;&#32;&#43;&#32;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"128\" style=\"vertical-align: -2px;\" \/>.)  Discuss how taking another path to reach the same point might help to overcome an obstacle blocking you other path.\n  <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-24\">\n<p id=\"eip-157\">\n    30.8 m, 35.8 west of north\n  <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1862376\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1862377\">\n<p id=\"import-auto-id1165298883937\">You drive <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c37d34a48795e643632fe98729f2a04d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/> in a straight line in a direction <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a6b3363bd2b10d8b0d6ae9c827e88abd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>  east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1629683\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1629684\">\n<p id=\"import-auto-id1165298883936\">Do <a href=\"#eip-287\" class=\"autogenerated-content\">(Figure)<\/a> again using analytical techniques and change the second leg of the walk to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1dc283ba655246eeea95717022aae476_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#46;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/> straight south. (This is equivalent to subtracting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> \u2014that is, finding<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d6b29f95d55c72c899a87e65d13f785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;&#125;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#32;&#45;&#32;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"85\" style=\"vertical-align: -1px;\" \/>) (b) Repeat again, but now you first walk <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-71d1aeabf35fc21a192a89e99ccf4153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/> north and then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fed4a83a9b4249854cfedbe5d394bf21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/> east. (This is equivalent to subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> \u2014that is, to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dc5d9e7ec16c7b091708071fc5627848_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"89\" style=\"vertical-align: -2px;\" \/>. Is that consistent with your result?)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1822911\">\n<p id=\"import-auto-id1165298883928\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0f1ea28c2afa4ec40d08c02584376944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d3fb0c6391f024571ae6efd520ace3fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#50;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> south of west<\/p>\n<p id=\"import-auto-id1165298676944\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0f1ea28c2afa4ec40d08c02584376944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d3fb0c6391f024571ae6efd520ace3fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#50;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> north of east<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1956316\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1956317\">\n<p id=\"import-auto-id1165298676945\">A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0672dc4d6f240c7ac13237d08e04908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> in <a href=\"#eip-id3165265\" class=\"autogenerated-content\">(Figure)<\/a>. She then correctly calculates the length and orientation of the third side <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ff48f7d5b2ba173b754abd52a3142c27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>. What is her result?<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id3165265\"><span data-type=\"media\" id=\"import-auto-id1165296607394\" data-alt=\"In the given figure the sides of a triangular piece of land are shown in vector form. West corner is at origin. A vector starts from the origin towards south east direction and makes an angle twenty-one degrees with the horizontal. Then from the head of this vector another vector B making an angle eleven degrees with the vertical is drawn upwards. Then another vector C from the head of the vector B to the tail of the initial vector is drawn. The length and orientation of side C is indicated as unknown, represented by a question mark.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_11a.jpg\" data-media-type=\"image\/wmf\" alt=\"In the given figure the sides of a triangular piece of land are shown in vector form. West corner is at origin. A vector starts from the origin towards south east direction and makes an angle twenty-one degrees with the horizontal. Then from the head of this vector another vector B making an angle eleven degrees with the vertical is drawn upwards. Then another vector C from the head of the vector B to the tail of the initial vector is drawn. The length and orientation of side C is indicated as unknown, represented by a question mark.\" width=\"250\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1874820\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1874821\">\n<p id=\"import-auto-id1165298676947\">You fly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-207acc8e73dbcedef2356c40ae289433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/> in a straight line in still air in the direction <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9b2afe3b0ff9e009a6b01b7febf56a80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> south of west. (a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.) (b) Find the distances you would have to fly first in a direction <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea24dabd3bfb3eb04c56478b6c973443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/> south of west and then in a direction <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea24dabd3bfb3eb04c56478b6c973443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/> west of north. These are the components of the displacement along a different set of axes\u2014one rotated <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce5319ee2d0e548b28a349b0f43ba034_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1797961\">\n<p id=\"import-auto-id1165298676948\">18.4 km south, then 26.2 km west(b) 31.5 km at<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea24dabd3bfb3eb04c56478b6c973443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/> south of west, then 5.56 km at<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea24dabd3bfb3eb04c56478b6c973443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/> west of north<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-379\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-201\">\n<p id=\"eip-222\">A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1ebf3e1b68a697e605007d0d8c74676f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/><br \/>\n <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dcc222f316e4ce548d7a695896b876e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6986798818c2f4e53663aad2275601f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> in <a href=\"#import-auto-id1165298543237\" class=\"autogenerated-content\">(Figure)<\/a>, and then correctly calculates the length and orientation of the fourth side <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82bccb3b7358fc3fe4ceea59a362e84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/>.<br \/>\n  What is his result?<\/p>\n<p id=\"eip-68\">\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298543237\"><span data-type=\"media\" id=\"import-auto-id1165298608985\" data-alt=\"A quadrilateral with sides A, B, C, and D. A begins at the end of D and is 4 point seven zero kilometers  at an angle of 7 point 5 degrees south of west. B begins at the end of A and is 2 point four eight kilometers in a direction sixteen degrees west of north. C begins at the end of B and is 3 point zero 2 kilometers in a direction nineteen degrees north of west. D begins at the end of C and runs distance and direction that must be calculated\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_12a.jpg\" data-media-type=\"image\/wmf\" alt=\"A quadrilateral with sides A, B, C, and D. A begins at the end of D and is 4 point seven zero kilometers  at an angle of 7 point 5 degrees south of west. B begins at the end of A and is 2 point four eight kilometers in a direction sixteen degrees west of north. C begins at the end of B and is 3 point zero 2 kilometers in a direction nineteen degrees north of west. D begins at the end of C and runs distance and direction that must be calculated\" width=\"300\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1849688\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1849689\">\n<p id=\"import-auto-id11652986769482\">In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-00f8aa92c09d800855e067263a6061ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea24dabd3bfb3eb04c56478b6c973443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/><\/p>\n<p> north of west; then <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1cb23ded05c34e5e97e0c036ed25743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f4e5fcc3de69cf7b1cd8fc6a432fbc76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> south of east; then <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31b23eccbad50bd374749f17ba99af07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#51;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"59\" style=\"vertical-align: -1px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae7fd0c7197b3a1e3d0934559354abc5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> south of west; then <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-75af5c3cac0184ced1ef4ac4cb6d1397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/> straight east; then <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d468f701a48044a87a548dc66f884cc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#55;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"59\" style=\"vertical-align: -1px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d4d6da9dbec773c0007154ba881bcf43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#48;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> east of north; then <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5ee2959e4b065685758ecb8dbb8e5298_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-126e879e4d067489344f66a05a2f0a02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> south of west; and finally <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc25738d5ad9a1324bdbd60cec0bb148_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-be018010a1d6022abdecae0d8b04959e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> north of east. What is his final position relative to the island?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1930787\">\n<p id=\"import-auto-id11652986769483\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2680d856beb2a7b7a6e80a732a416211_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e0d331d0fb2151192c136f094c62ef82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> south of east<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1955218\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1955219\">\n<p id=\"import-auto-id11652986769484\">Suppose a pilot flies <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a485207cf62f276800d695d1a8519ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/> in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f58822796ed6c341868803a248de619_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> north of east and then flies <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c61f1f7c535edec02afb8c5c4f6072e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/> in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/> north of east as shown in <a href=\"#import-auto-id1165298708571\" class=\"autogenerated-content\">(Figure)<\/a>. Find her total distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> from the starting point and the direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> of the straight-line path to the final position. Discuss qualitatively how this flight would be altered by a wind from the north and how the effect of the wind would depend on both wind speed and the speed of the plane relative to the air mass.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298708571\"><span data-type=\"media\" id=\"import-auto-id1165298708572\" data-alt=\"A triangle  defined by vectors A, B, and R. A begins at the origin and run forty kilometers in a direction sixty degrees north of east. B begins at the end of A and runs thirty kilometers in a direction fifteen degrees north of east. R is the resultant vector and runs from the origin (the beginning of A) to the end of B for a distance and in a direction theta that need to be calculated.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_03_13a.jpg\" data-media-type=\"image\/wmf\" alt=\"A triangle  defined by vectors A, B, and R. A begins at the origin and run forty kilometers in a direction sixty degrees north of east. B begins at the end of A and runs thirty kilometers in a direction fifteen degrees north of east. R is the resultant vector and runs from the origin (the beginning of A) to the end of B for a distance and in a direction theta that need to be calculated.\" width=\"275\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1165299001085\">\n<dt>analytical method<\/dt>\n<dd id=\"fs-id1544968\">the method of determining the magnitude and direction of a resultant vector using the Pythagorean theorem and trigonometric identities<\/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-194","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":145,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/194\/revisions"}],"predecessor-version":[{"id":195,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/194\/revisions\/195"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/145"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/194\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=194"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=194"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}