{"id":179,"date":"2017-10-27T16:29:05","date_gmt":"2017-10-27T16:29:05","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/vector-addition-and-subtraction-graphical-methods\/"},"modified":"2017-11-08T03:23:57","modified_gmt":"2017-11-08T03:23:57","slug":"vector-addition-and-subtraction-graphical-methods","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/vector-addition-and-subtraction-graphical-methods\/","title":{"raw":"Vector Addition and Subtraction: Graphical Methods","rendered":"Vector Addition and Subtraction: Graphical 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, subtraction, and multiplication.<\/li>\n<li>Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296227310\">\n<div class=\"bc-figcaption figcaption\">Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai\u2019i to Moloka\u2019i has a number of legs, or journey segments. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. (credit: US Geological Survey)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298679996\" data-alt=\"Some Hawaiian Islands like Kauai Oahu, Molokai, Lanai, Maui, Kahoolawe, and Hawaii are shown. On the scale map of Hawaiian Islands the path of a journey is shown moving from Hawaii to Molokai. The path of the journey is turning at different angles and finally reaching its destination. The displacement of the journey is shown with the help of a straight line connecting its starting point and the destination.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_00a.jpg\" data-media-type=\"image\/png\" alt=\"Some Hawaiian Islands like Kauai Oahu, Molokai, Lanai, Maui, Kahoolawe, and Hawaii are shown. On the scale map of Hawaiian Islands the path of a journey is shown moving from Hawaii to Molokai. The path of the journey is turning at different angles and finally reaching its destination. The displacement of the journey is shown with the help of a straight line connecting its starting point and the destination.\" width=\"300\"><\/span><\/p><\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165296240221\">\n<h1 data-type=\"title\">Vectors in Two Dimensions<\/h1>\n<p id=\"import-auto-id1165298552138\">A <span data-type=\"term\" id=\"import-auto-id1165296389850\">vector<\/span> is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector\u2019s magnitude and pointing in the direction of the vector.<\/p>\n<p id=\"import-auto-id1165298918938\"><a href=\"#import-auto-id1165298666909\" class=\"autogenerated-content\">(Figure)<\/a> shows such a <em data-effect=\"italics\">graphical representation of a vector<\/em>, using as an example the total displacement for the person walking in a city considered in <a href=\"\/contents\/21d0e217-d50f-4901-af75-905e738eb4c4@4\">Kinematics in Two Dimensions: An Introduction<\/a>. We shall use the notation that a boldface symbol, such as [latex]\\text{D}[\/latex], stands for a vector. Its magnitude is represented by the symbol in italics, [latex]D[\/latex], and its direction by [latex]\\theta [\/latex].<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165296218458\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Vectors in this Text<\/div>\n<p>In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector [latex]\\text{F}[\/latex], which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as [latex]F[\/latex], and the direction of the variable will be given by an angle [latex]\\theta [\/latex].<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298666909\">\n<div class=\"bc-figcaption figcaption\">A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle [latex]\\text{29}\\text{.1\u00ba}[\/latex] north of east.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298713446\" data-alt=\"A graph is shown. On the axes the scale is set to one block is equal to one unit. A helicopter starts moving from the origin at an angle of twenty nine point one degrees above the x axis. The current position of the helicopter is ten point three blocks along its line of motion. The destination of the helicopter is the point which is nine blocks in the positive x direction and five blocks in the positive y direction. The positive direction of the x axis is east and the positive direction of the y axis is north.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_01.jpg\" data-media-type=\"image\/jpg\" alt=\"A graph is shown. On the axes the scale is set to one block is equal to one unit. A helicopter starts moving from the origin at an angle of twenty nine point one degrees above the x axis. The current position of the helicopter is ten point three blocks along its line of motion. The destination of the helicopter is the point which is nine blocks in the positive x direction and five blocks in the positive y direction. The positive direction of the x axis is east and the positive direction of the y axis is north.\" width=\"325\"><\/span><\/p><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298918248\">\n<div class=\"bc-figcaption figcaption\">To describe the resultant vector for the person walking in a city considered in <a href=\"#import-auto-id1165298666909\" class=\"autogenerated-content\">(Figure)<\/a> graphically, draw an arrow to represent the total displacement vector [latex]\\text{D}[\/latex]. Using a protractor, draw a line at an angle [latex]\\theta [\/latex] relative to the east-west axis. The length [latex]D[\/latex] of the arrow is proportional to the vector\u2019s magnitude and is measured along the line with a ruler. In this example, the magnitude [latex]D[\/latex] of the vector is 10.3 units, and the direction [latex]\\theta [\/latex] is [latex]29.1\u00ba[\/latex] north of east.\n<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296263736\" data-alt=\"On a graph a vector is shown. It is inclined at an angle theta equal to twenty nine point one degrees above the positive x axis. A protractor is shown to the right of the x axis to measure the angle. A ruler is also shown parallel to the vector to measure its length. The ruler shows that the length of the vector is ten point three units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"On a graph a vector is shown. It is inclined at an angle theta equal to twenty nine point one degrees above the positive x axis. A protractor is shown to the right of the x axis to measure the angle. A ruler is also shown parallel to the vector to measure its length. The ruler shows that the length of the vector is ten point three units.\" height=\"250\"><\/span><\/p><\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298995028\">\n<h1 data-type=\"title\">Vector Addition: Head-to-Tail Method<\/h1>\n<p id=\"import-auto-id1165298553339\">The <span data-type=\"term\" id=\"import-auto-id1165298552505\">head-to-tail method<\/span> is a graphical way to add vectors, described in <a href=\"#import-auto-id1165298643218\" class=\"autogenerated-content\">(Figure)<\/a> below and in the steps following. The <span data-type=\"term\">tail<\/span> of the vector is the starting point of the vector, and the <span data-type=\"term\" id=\"import-auto-id1165298982372\">head<\/span> (or tip) of a vector is the final, pointed end of the arrow.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298643218\">\n<div class=\"bc-figcaption figcaption\"><strong>Head-to-Tail Method:<\/strong> The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in <a href=\"#import-auto-id1165298666909\" class=\"autogenerated-content\">(Figure)<\/a>.  (a) Draw a vector representing the displacement to the east. (b) Draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first, east-pointing vector. (c) Draw a line from the tail of the east-pointing vector to the head of the north-pointing vector to form the sum or <span data-type=\"term\">resultant vector<\/span> [latex]\\text{D}[\/latex]. The length of the arrow [latex]\\text{D}[\/latex] is proportional to the vector\u2019s magnitude and is measured to be 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) [latex]\\theta [\/latex] is measured with a protractor to be [latex]\\text{29}\\text{.}1\u00ba[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298533716\" data-alt=\"In part a, a vector of magnitude of nine units and making an angle of theta is equal to zero degrees is drawn from the origin and along the positive direction of x axis. In part b a vector of magnitude of nine units and making an angle of theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical arrow from the head of the horizontal arrow is drawn. In part c a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vectorD.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_03.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a, a vector of magnitude of nine units and making an angle of theta is equal to zero degrees is drawn from the origin and along the positive direction of x axis. In part b a vector of magnitude of nine units and making an angle of theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical arrow from the head of the horizontal arrow is drawn. In part c a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vectorD.\" width=\"500\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165296543683\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 1.<\/em><\/strong><em data-effect=\"italics\"><em data-effect=\"italics\">Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor<\/em><\/em>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298876451\"><span data-type=\"media\" id=\"import-auto-id1165298800021\" data-alt=\"In part a, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis.\" height=\"200\"><\/span><\/div>\n<p id=\"import-auto-id1165298805929\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 2.<\/em><\/strong> Now draw an arrow to represent the second vector (5 blocks to the north). <em data-effect=\"italics\">Place the tail of the second vector at the head of the first vector<\/em>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298818267\"><span data-type=\"media\" id=\"import-auto-id1165298672083\" data-alt=\"In part b, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical vector from the head of the horizontal vector is drawn.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part b, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical vector from the head of the horizontal vector is drawn.\" height=\"200\"><\/span><\/div>\n<p id=\"import-auto-id1165296690060\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 3.<\/em><\/strong><em data-effect=\"italics\">If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail<\/em>.<\/p>\n<p id=\"import-auto-id1165298943938\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 4.<\/em><\/strong><em data-effect=\"italics\">Draw an arrow from the tail of the first vector to the head of the last vector<\/em>. This is the <span data-type=\"term\" id=\"import-auto-id1165296311722\">resultant<\/span>, or the sum, of the other vectors.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165299000967\"><span data-type=\"media\" id=\"import-auto-id1165298799464\" data-alt=\"In part c, a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of the x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vector D.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part c, a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of the x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vector D.\" width=\"200\"><\/span><\/div>\n<p id=\"import-auto-id1165298478340\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 5.<\/em><\/strong> To get the <span data-type=\"term\" id=\"import-auto-id1165298794109\">magnitude<\/span> of the resultant, <em data-effect=\"italics\">measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.)<\/em><\/p>\n<p id=\"import-auto-id1165298643122\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 6. <\/em><\/strong>To get the <span data-type=\"term\" id=\"import-auto-id1165298932041\">direction<\/span> of the resultant, <em data-effect=\"italics\"><em data-effect=\"italics\">measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.)<\/em><\/em><\/p>\n<p id=\"import-auto-id1165298452161\">The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296298332\">\n<div data-type=\"title\" class=\"title\">Adding Vectors Graphically Using the Head-to-Tail Method: A Woman Takes a Walk<\/div>\n<p id=\"import-auto-id1165298774683\">Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction [latex]\\text{49.0\u00ba}[\/latex] north of east. Then, she walks 23.0 m heading [latex]\\text{15.0\u00ba}[\/latex] north of east. Finally, she turns and walks 32.0 m in a direction 68.0\u00b0 south of east.<\/p>\n<p id=\"import-auto-id1165296336458\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1165298723286\">Represent each displacement vector graphically with an arrow, labeling the first [latex]\\text{A}[\/latex], the second [latex]\\text{B}[\/latex], and the third [latex]\\text{C}[\/latex], making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted [latex]\\mathbf{\\text{R}}[\/latex].<\/p>\n<p id=\"import-auto-id1165298842510\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165298879632\">(1) Draw the three displacement vectors.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296232338\"><span data-type=\"media\" id=\"import-auto-id1165298941346\" data-alt=\"On the graph a vector of magnitude twenty three meters and inclined above the x axis at an angle theta-b equal to fifteen degrees is shown. This vector is labeled as B.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_08.jpg\" data-media-type=\"image\/jpg\" alt=\"On the graph a vector of magnitude twenty three meters and inclined above the x axis at an angle theta-b equal to fifteen degrees is shown. This vector is labeled as B.\" width=\"500\"><\/span><\/div>\n<p id=\"import-auto-id1165298699961\">(2) Place the vectors head to tail retaining both their initial magnitude and direction.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298788198\"><span data-type=\"media\" id=\"import-auto-id1165296306377\" data-alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_09.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis.\" width=\"250\"><\/span><\/div>\n<p id=\"import-auto-id1165298517765\">(3) Draw the resultant vector, [latex]\\text{R}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298786300\"><span data-type=\"media\" id=\"import-auto-id1165298835331\" data-alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_10.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector.\" width=\"250\"><\/span><\/div>\n<p id=\"import-auto-id1165296319738\">(4) Use a ruler to measure the magnitude of [latex]\\mathbf{\\text{R}}[\/latex], and a protractor to measure the direction of [latex]\\text{R}[\/latex]. While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298931707\"><span data-type=\"media\" id=\"import-auto-id1165296287125\" data-alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty meter and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector. A ruler is placed along the vector R to measure it. Also there is a protractor to measure the angle.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_11a.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty meter and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector. A ruler is placed along the vector R to measure it. Also there is a protractor to measure the angle.\" width=\"220\"><\/span><\/div>\n<p id=\"import-auto-id1165298598693\">In this case, the total displacement [latex]\\mathbf{\\text{R}}[\/latex] is seen to have a magnitude of 50.0 m and to lie in a direction [latex]7.0\u00ba[\/latex] south of east. By using its magnitude and direction, this vector can be expressed as [latex]R=\\text{50.0 m}[\/latex] and [latex]\\theta =7\\text{.}\\text{0\u00ba}[\/latex] south of east.<\/p>\n<p id=\"import-auto-id1165298639141\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id1165296455316\">      The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in <a href=\"#import-auto-id1165298931858\" class=\"autogenerated-content\">(Figure)<\/a> and we will still get the same solution.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298931858\"><span data-type=\"media\" id=\"import-auto-id1165296377152\" data-alt=\"In this figure a vector C with a negative slope is drawn from the origin. Then from the head of the vector C another vector A with positive slope is drawn and then another vector B with negative slope from the head of the vector A is drawn. From the tail of the vector C a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector B. The vector R is known as the resultant vector.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_12.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector C with a negative slope is drawn from the origin. Then from the head of the vector C another vector A with positive slope is drawn and then another vector B with negative slope from the head of the vector A is drawn. From the tail of the vector C a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector B. The vector R is known as the resultant vector.\" width=\"275\"><\/span><\/div>\n<p id=\"import-auto-id1165296613647\">Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is <span data-type=\"term\">commutative<\/span>. Vectors can be added in any order.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-376\">[latex]\\mathbf{\\text{A}}+\\mathbf{\\text{B}}=\\mathbf{\\text{B}}+\\mathbf{\\text{A}}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1165298670064\">(This is true for the addition of ordinary numbers as well\u2014you get the same result whether you add [latex]\\mathbf{\\text{2}}+\\mathbf{\\text{3}}[\/latex]<br>\n or<\/p>\n<p>[latex]\\mathbf{\\text{3}}+\\mathbf{\\text{2}}[\/latex], for example).<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298779158\">\n<h1 data-type=\"title\">Vector Subtraction<\/h1>\n<p id=\"import-auto-id1165298786530\">Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract [latex]\\mathbf{\\text{B}}[\/latex] from <\/p>\n[latex]\\mathbf{\\text{A}}[\/latex]\n<p>, written [latex]\\mathbf{\\text{A}}\u2013\\mathbf{\\text{B}}[\/latex]<\/p>\n<p>, we must first define what we mean by subtraction. The <em data-effect=\"italics\">negative<\/em> of a vector [latex]\\mathbf{\\text{B}}[\/latex]<\/p>\n<p>is defined to be [latex]\\mathbf{\\text{\u2013B}}[\/latex]; that is, graphically <em data-effect=\"italics\">the negative of any vector has the same magnitude but the opposite direction<\/em>, as shown in <a href=\"#import-auto-id1165298692950\" class=\"autogenerated-content\">(Figure)<\/a>. In other words, [latex]\\mathbf{\\text{B}}[\/latex] has the same length as [latex]\\mathbf{\\text{\u2013B}}[\/latex], but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298692950\">\n<div class=\"bc-figcaption figcaption\">The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So [latex]\\mathbf{\\text{B}}[\/latex] is the negative of [latex]\\mathbf{\\text{\u2013B}}[\/latex]; it has the same length but opposite direction.      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296266911\" data-alt=\"Two vectors are shown. One of the vectors is labeled as vector   in north east direction. The other vector is of the same magnitude and is in the opposite direction to that of vector B. This vector is denoted as negative B.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_13a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors are shown. One of the vectors is labeled as vector   in north east direction. The other vector is of the same magnitude and is in the opposite direction to that of vector B. This vector is denoted as negative B.\" height=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298788891\">The <em data-effect=\"italics\"><em data-effect=\"italics\">subtraction<\/em><\/em> of vector [latex]\\mathbf{\\text{B}}[\/latex] from vector [latex]\\mathbf{\\text{A}}[\/latex] is then simply defined to be the addition of [latex]\\mathbf{\\text{\u2013B}}[\/latex] to [latex]\\mathbf{\\text{A}}[\/latex]. Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-454\">[latex]\\text{A&nbsp;\u2013&nbsp;B&nbsp;=&nbsp;A&nbsp;+&nbsp;}\\left(\\text{\u2013B}\\right)\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1165298555304\">This is analogous to the subtraction of scalars (where, for example, [latex]\\text{5&nbsp;\u2013&nbsp;2&nbsp;=&nbsp;5&nbsp;+&nbsp;}\\left(\\text{\u20132}\\right)[\/latex]). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296679497\">\n<div data-type=\"title\" class=\"title\">Subtracting Vectors Graphically: A Woman Sailing a Boat<\/div>\n<p id=\"import-auto-id1165298586222\">A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction [latex]\\text{66.0\u00ba}[\/latex] north of east from her current location, and then travel 30.0 m in a direction [latex]\\text{112\u00ba}[\/latex] north of east (or [latex]\\text{22.0\u00ba}[\/latex] west of north). If the woman makes a mistake and travels in the <em data-effect=\"italics\"><em data-effect=\"italics\">opposite<\/em><\/em> direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296408744\"><span data-type=\"media\" id=\"import-auto-id1165296232464\" data-alt=\"A vector of magnitude twenty seven point five meters is shown. It is inclined to the horizontal at an angle of sixty six degrees. Another vector of magnitude thirty point zero meters is shown. It is inclined to the horizontal at an angle of one hundred and twelve degrees.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_14.jpg\" data-media-type=\"image\/jpg\" alt=\"A vector of magnitude twenty seven point five meters is shown. It is inclined to the horizontal at an angle of sixty six degrees. Another vector of magnitude thirty point zero meters is shown. It is inclined to the horizontal at an angle of one hundred and twelve degrees.\" width=\"450\"><\/span><\/div>\n<p id=\"import-auto-id1165298644722\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1165298832023\">We can represent the first leg of the trip with a vector [latex]\\mathbf{\\text{A}}[\/latex], and the second leg of the trip with a vector <\/p>\n<p>[latex]\\mathbf{\\text{B}}[\/latex]. The dock is located at a location [latex]\\mathbf{\\text{A}}+\\mathbf{\\text{B}}[\/latex]. If the woman mistakenly travels in the <em data-effect=\"italics\">opposite<\/em> direction for the second leg of the journey, she will travel a distance [latex]B[\/latex]  (30.0 m) in the direction [latex]180\u00ba\u2013112\u00ba=68\u00ba[\/latex]  south of east. We represent this as [latex]\\mathbf{\\text{\u2013B}}[\/latex], as shown below. The vector [latex]\\mathbf{\\text{\u2013B}}[\/latex] has the same magnitude as [latex]\\mathbf{\\text{B}}[\/latex] but is in the opposite direction. Thus, she will end up at a location [latex]\\mathbf{\\text{A}}+\\left(\\mathbf{\\text{\u2013B}}\\right)[\/latex], or [latex]\\mathbf{\\text{A}}\u2013\\mathbf{\\text{B}}[\/latex]. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296408745\"><span data-type=\"media\" id=\"import-auto-id1165296232465\" data-alt=\"A vector labeled negative B is inclined at an angle of sixty-eight degrees below a horizontal line. A dotted line in the reverse direction inclined at one hundred and twelve degrees above the horizontal line is also shown.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_15a.jpg\" data-media-type=\"image\/jpg\" alt=\"A vector labeled negative B is inclined at an angle of sixty-eight degrees below a horizontal line. A dotted line in the reverse direction inclined at one hundred and twelve degrees above the horizontal line is also shown.\" width=\"200\"><\/span><\/div>\n<p id=\"import-auto-id1165298473863\">We will perform vector addition to compare the location of the dock, [latex]\\text{A&nbsp;}\\text{+&nbsp;}\\mathbf{B}[\/latex], with the location at which the woman mistakenly arrives, [latex]\\text{A&nbsp;+&nbsp;}\\left(\\text{\u2013B}\\right)[\/latex].<\/p>\n<p id=\"import-auto-id1165298943436\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296259545\">(1) To determine the location at which the woman arrives by accident, draw vectors [latex]\\mathbf{\\text{A}}[\/latex] and [latex]\\mathbf{\\text{\u2013B}}[\/latex].<\/p>\n<p id=\"import-auto-id1165298618212\">(2) Place the vectors head to tail.<\/p>\n<p id=\"import-auto-id1165296239091\">(3) Draw the resultant vector [latex]\\mathbf{R}[\/latex].<\/p>\n<p id=\"import-auto-id1165298732001\">(4) Use a ruler and protractor to measure the magnitude and direction of [latex]\\mathbf{R}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298476927\"><span data-type=\"media\" id=\"import-auto-id1165298786656\" data-alt=\"Vectors A and negative B are connected in head to tail method. Vector A is inclined with horizontal with positive slope and vector negative B with a negative slope. The resultant of these two vectors is shown as a vector R from tail of A to the head of negative B. The length of the resultant is twenty three point zero meters and has a negative slope of seven point five degrees.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_16a.jpg\" data-media-type=\"image\/jpg\" alt=\"Vectors A and negative B are connected in head to tail method. Vector A is inclined with horizontal with positive slope and vector negative B with a negative slope. The resultant of these two vectors is shown as a vector R from tail of A to the head of negative B. The length of the resultant is twenty three point zero meters and has a negative slope of seven point five degrees.\" width=\"300\"><\/span><\/div>\n<p id=\"import-auto-id1165298710088\">In this case, [latex]R=\\text{23}\\text{.}\\text{0 m}[\/latex]<br>\n    and<br>\n[latex]\\theta =7\\text{.}\\text{5\u00ba}[\/latex]<br>\n    south of east.<\/p>\n<p id=\"import-auto-id1165298996239\">(5) To determine the location of the dock, we repeat this method to add vectors [latex]\\mathbf{\\text{A}}[\/latex] and [latex]\\mathbf{\\text{B}}[\/latex]. We obtain the resultant vector [latex]\\mathbf{\\text{R}}\\text{'}[\/latex]:<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296298190\"><span data-type=\"media\" id=\"import-auto-id1165296574919\" data-alt=\"A vector A inclined at sixty six degrees with horizontal is shown. From the head of this vector another vector B is started. Vector B is inclined at one hundred and twelve degrees with the horizontal. Another vector labeled as R prime from the tail of vector A to the head of vector B is drawn. The length of this vector is fifty two point nine meters and its inclination with the horizontal is shown as ninety point one degrees. Vector R prime is equal to the sum of vectors A and B.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_17a.jpg\" data-media-type=\"image\/jpg\" alt=\"A vector A inclined at sixty six degrees with horizontal is shown. From the head of this vector another vector B is started. Vector B is inclined at one hundred and twelve degrees with the horizontal. Another vector labeled as R prime from the tail of vector A to the head of vector B is drawn. The length of this vector is fifty two point nine meters and its inclination with the horizontal is shown as ninety point one degrees. Vector R prime is equal to the sum of vectors A and B.\" width=\"250\"><\/span><\/div>\n<p id=\"import-auto-id1165298652611\">In this case [latex]R\\text{&nbsp;=&nbsp;52.9 m}[\/latex]<br>\n    and [latex]\\theta =\\text{90.1\u00ba}[\/latex]<br>\n    &nbsp;north&nbsp;of&nbsp;east.<\/p>\n<p id=\"import-auto-id1165296220926\">We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.<\/p>\n<p id=\"import-auto-id1165298719642\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id1165296245111\">Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298652611\">\n<h1 data-type=\"title\">Multiplication of Vectors and Scalars<\/h1>\n<p id=\"import-auto-id1165298868552\">If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk [latex]\\text{3&nbsp;}\u00d7\\text{&nbsp;27}\\text{.}\\text{5 m}[\/latex], or 82.5 m, in a direction [latex]\\text{66}\\text{.}0\\text{\u00ba}[\/latex] north of east. This is an example of multiplying a vector by a positive <span data-type=\"term\" id=\"import-auto-id1165296219603\">scalar<\/span>. Notice that the magnitude changes, but the direction stays the same.<\/p>\n<p id=\"import-auto-id1165298838383\">If the scalar is negative, then multiplying a vector by it changes the vector\u2019s magnitude and gives the new vector the <em data-effect=\"italics\"><em data-effect=\"italics\">opposite<\/em><\/em> direction. For example, if you multiply by \u20132, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector [latex]\\mathbf{A}[\/latex] is multiplied by a scalar [latex]c[\/latex],<\/p>\n<ul id=\"fs-id1165298531076\">\n<li id=\"import-auto-id1165298651742\">the magnitude of the vector becomes the absolute value of [latex]c[\/latex][latex]A[\/latex],<\/li>\n<li id=\"import-auto-id1165298881521\">if [latex]c[\/latex] is positive, the direction of the vector does not change,<\/li>\n<li id=\"import-auto-id1165298455414\">if [latex]c[\/latex] is negative, the direction is reversed.<\/li>\n<\/ul>\n<p id=\"import-auto-id1165298960255\">In our case, [latex]c=3[\/latex] and [latex]A=27.5 m[\/latex]. Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1\/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298819725\">\n<h1 data-type=\"title\">Resolving a Vector into Components<\/h1>\n<p id=\"import-auto-id1165298553346\">In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular <span data-type=\"term\" id=\"import-auto-id1165298555736\">components <\/span>of a single vector, for example the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-<em data-effect=\"italics\"> and<\/em> <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components, or the north-south and east-west components.<\/p>\n<p id=\"import-auto-id1165298883158\">For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction [latex]\\text{29}\\text{.0\u00ba}[\/latex] north of east and want to find out how many blocks east and north had to be walked. This method is called <em data-effect=\"italics\"><em data-effect=\"italics\">finding the components (or parts)<\/em><\/em> of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in <a href=\"\/contents\/69062f44-56d2-4111-88ff-f599727c4ed1@12\">Projectile Motion<\/a>, and much more when we cover <strong>forces<\/strong>  in <a href=\"\/contents\/02f52a02-2484-4ccb-bbd4-3c94edaa8e09@4\">Dynamics: Newton\u2019s Laws of Motion<\/a>. Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in <a href=\"\/contents\/b9739bfd-dc9d-4f0a-b037-dc22884d30f3@10\">Vector Addition and Subtraction: Analytical Methods<\/a> are ideal for finding vector components.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Maze Game<\/div>\n<p id=\"eip-id2002304\">Learn about position, velocity, and acceleration in the \"Arena of Pain\". Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1434453\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/589973fcfa456bcabda2138e415688acf6b73e6f\/maze-game_en.jar\">Maze Game<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_3.2\" data-alt=\"\"><a href=\"\/resources\/589973fcfa456bcabda2138e415688acf6b73e6f\/maze-game_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-id1165298622440\">\n<h1 data-type=\"title\">Summary<\/h1>\n<ul id=\"fs-id1165298751188\">\n<li id=\"import-auto-id1165296253334\">The <strong>graphical method of adding vectors<\/strong> [latex]\\mathbf{A}[\/latex] and [latex]\\mathbf{B}[\/latex] involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector\n<p>[latex]\\mathbf{R}[\/latex] is defined such that <\/p>\n<p>[latex]\\mathbf{\\text{A}}+\\mathbf{\\text{B}}=\\mathbf{\\text{R}}[\/latex]. The magnitude and direction of [latex]\\mathbf{R}[\/latex] are then determined with a ruler and protractor, respectively.<\/p><\/li>\n<li id=\"import-auto-id1165298573640\">The <strong>graphical method of subtracting vector <\/strong> [latex]\\mathbf{B}[\/latex] from [latex]\\mathbf{A}[\/latex] involves adding the opposite of vector [latex]\\mathbf{B}[\/latex], which is defined as [latex]-\\mathbf{B}[\/latex]. In this case, [latex]\\text{A}\u2013\\mathbf{\\text{B}}=\\mathbf{\\text{A}}+\\left(\\text{\u2013B}\\right)=\\text{R}[\/latex]. Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector [latex]\\mathbf{R}[\/latex].<\/li>\n<li id=\"import-auto-id1165296680072\">Addition of vectors is <span data-type=\"term\" id=\"import-auto-id1165296680069\">commutative<\/span> such that [latex]\\mathbf{\\text{A}}+\\mathbf{\\text{B}}=\\mathbf{\\text{B}}+\\mathbf{\\text{A}}[\/latex] .<\/li>\n<li id=\"import-auto-id1165296269519\">The <span data-type=\"term\" id=\"import-auto-id1165298982089\">head-to-tail method<\/span> of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.<\/li>\n<li id=\"import-auto-id1165298819524\">If a vector [latex]\\mathbf{A}[\/latex] is multiplied by a scalar quantity [latex]c[\/latex], the magnitude of the product is given by [latex]\\text{cA}[\/latex]. If [latex]c[\/latex] is positive, the direction of the product points in the same direction as [latex]\\mathbf{A}[\/latex]; if [latex]c[\/latex] is negative, the direction of the product points in the opposite direction as [latex]\\mathbf{A}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1165299003649\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298732767\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298730553\">\n<p id=\"import-auto-id1165298740934\">Which of the following is a vector: a person\u2019s height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this book, the Earth\u2019s population, the acceleration of gravity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298775836\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298806849\">\n<p id=\"import-auto-id1165298595710\">Give a specific example of a vector, stating its magnitude, units, and direction.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298783616\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298725379\">\n<p id=\"import-auto-id1165296264337\">What do vectors and scalars have in common? How do they differ?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296576944\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298861181\">\n<p id=\"import-auto-id1165296579324\">Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298840401\"><span data-type=\"media\" id=\"import-auto-id1165298840402\" data-alt=\"At the southwest corner of the figure is a cabin and in the northeast corner is a lake. A vector S with a length five point zero kilometers connects the cabin to the lake at an angle of 40 degrees north of east. Two winding paths labeled Path 1 and Path 2 represent the routes travelled from the cabin to the lake.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_18a.jpg\" data-media-type=\"image\/wmf\" alt=\"At the southwest corner of the figure is a cabin and in the northeast corner is a lake. A vector S with a length five point zero kilometers connects the cabin to the lake at an angle of 40 degrees north of east. Two winding paths labeled Path 1 and Path 2 represent the routes travelled from the cabin to the lake.\" width=\"275\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298981819\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298981820\">\n<p id=\"import-auto-id1165296218026\">If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in <a href=\"#import-auto-id1165296384452\" class=\"autogenerated-content\">(Figure)<\/a>. What other information would he need to get to Sacramento?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296384452\"><span data-type=\"media\" id=\"import-auto-id1165296384453\" 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.\" height=\"300\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298754450\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298754451\">\n<p id=\"import-auto-id1165298998383\">Suppose you take two steps [latex]\\mathbf{\\text{A}}[\/latex] and [latex]\\mathbf{\\text{B}}[\/latex]  (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point [latex]\\mathbf{\\text{A}}+\\mathbf{\\text{B}}[\/latex] the sum of the lengths of the two steps?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298761188\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298993266\">\n<p id=\"import-auto-id1165296716271\">Explain why it is not possible to add a scalar to a vector.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296242489\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298645243\">\n<p id=\"import-auto-id1165296408227\">If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1165298586130\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"import-auto-id1165298672665\"><strong data-effect=\"bold\">Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.<\/strong><\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298745593\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296363125\">\n<p id=\"import-auto-id1165298838476\">Find the following for path A in <a href=\"#import-auto-id1165298872310\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298872310\">\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-id1165298872312\" data-alt=\"A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meters each. The path of A extends to three blocks towards north and then one block towards east. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish.\"><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 meters each. The path of A extends to three blocks towards north and then one block towards east. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish.\" width=\"400\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296403819\">\n<p id=\"import-auto-id1165298835475\">(a) [latex]\\text{480 m}[\/latex]<\/p>\n<p id=\"import-auto-id1165296220703\">(b) [latex]\\text{379 m}[\/latex], [latex]\\text{18.4\u00ba}[\/latex] east of north<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298474424\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298770529\">\n<p id=\"import-auto-id1165298723075\">Find the following for path B in <a href=\"#import-auto-id1165298872310\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298867580\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296248676\">\n<p id=\"import-auto-id1165296365282\">Find the north and east components of the displacement for the hikers shown in <a href=\"#import-auto-id1165298840401\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296301840\">\n<p id=\"import-auto-id1165298650835\">north component 3.21 km, east component 3.83 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298536705\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296255759\">\n<p id=\"import-auto-id1165296243127\">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{\\text{A}}[\/latex] and [latex]\\mathbf{\\text{B}}[\/latex], as in <a href=\"#import-auto-id1165296241785\" class=\"autogenerated-content\">(Figure)<\/a>, then this problem asks you to find their sum [latex]\\mathbf{\\text{R}}=\\mathbf{\\text{A}}+\\mathbf{\\text{B}}[\/latex].)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296241785\">\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-id1165298744046\" data-alt=\"In this figure coordinate axes are shown. Vector A from the origin towards the negative of x axis is shown. From the head of the vector A another vector B is drawn towards the positive direction of y axis. The resultant R of these two vectors is shown as a vector from the tail of vector A to the head of vector B. This vector R is inclined at an angle theta with the negative x axis.\"><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 this figure coordinate axes are shown. Vector A from the origin towards the negative of x axis is shown. From the head of the vector A another vector B is drawn towards the positive direction of y axis. The resultant R of these two vectors is shown as a vector from the tail of vector A to the head of vector B. This vector R is inclined at an angle theta with the negative x axis.\" width=\"275\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298797729\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298797730\">\n<p id=\"import-auto-id1165298942186\">Suppose you first walk 12.0 m in a direction [latex]\\text{20\u00ba}[\/latex] west of north and then 20.0 m in a direction [latex]\\text{40.0\u00ba}[\/latex] south of west. 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-id1165296430663\" class=\"autogenerated-content\">(Figure)<\/a>, then this problem finds their sum [latex]\\text{R&nbsp;=&nbsp;A&nbsp;+&nbsp;B}[\/latex].)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296430663\"><span data-type=\"media\" id=\"import-auto-id1165296430664\" data-alt=\"In the given figure coordinates axes are shown. Vector A with tail at origin is inclined at an angle of twenty degrees with the positive direction of x axis. The magnitude of vector A is twelve meters. Another vector B is starts from the head of vector A and inclined at an angle of forty degrees with the horizontal. The resultant R of the vectors A and B is also drawn from the tail of vector A to the head of vector B. The inclination of vector R is theta with the horizontal.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_22a.jpg\" data-media-type=\"image\/wmf\" alt=\"In the given figure coordinates axes are shown. Vector A with tail at origin is inclined at an angle of twenty degrees with the positive direction of x axis. The magnitude of vector A is twelve meters. Another vector B is starts from the head of vector A and inclined at an angle of forty degrees with the horizontal. The resultant R of the vectors A and B is also drawn from the tail of vector A to the head of vector B. The inclination of vector R is theta with the horizontal.\" width=\"250\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298995751\">\n<p id=\"import-auto-id1165298542917\">[latex]\\text{19}\\text{.}\\text{5 m}[\/latex], [latex]4\\text{.}\\text{65\u00ba}[\/latex] south of west<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298849088\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296252134\">\n<p id=\"import-auto-id1165298727195\">Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg [latex]\\mathbf{B}[\/latex], which is 20.0 m in a direction exactly [latex]\\text{40\u00ba}[\/latex] south of west, and then leg [latex]\\mathbf{A}[\/latex], which is 12.0 m in a direction exactly [latex]\\text{20\u00ba}[\/latex] west of north. (This problem shows that [latex]\\mathbf{A}+\\mathbf{B}=\\mathbf{B}+\\mathbf{A}[\/latex].)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298560552\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296243866\">\n<p id=\"import-auto-id1165298727198\">(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction [latex]\\text{40.0\u00ba}[\/latex] north of east (which is equivalent to subtracting [latex]\\mathbf{\\text{B}}[\/latex] from [latex]\\mathbf{A}[\/latex] \u2014that is, to finding [latex]\\mathbf{\\text{R}}\\prime =\\mathbf{\\text{A}}-\\mathbf{\\text{B}}[\/latex]). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction [latex]\\text{40.0\u00ba}[\/latex] south of west and then 12.0 m in a direction [latex]\\text{20.0\u00ba}[\/latex] east of south (which is equivalent to subtracting <\/p>\n<p>[latex]\\mathbf{\\text{A}}[\/latex] from [latex]\\mathbf{\\text{B}}[\/latex] \u2014that is, to finding <\/p>\n<p>[latex]\\mathbf{\\text{R}}\\prime \\prime =\\mathbf{\\text{B}}-\\mathbf{\\text{A}}=-\\mathbf{\\text{R}}\\prime [\/latex]). Show that this is the case.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296252966\">\n<p id=\"import-auto-id1165296541089\">(a) [latex]\\text{26}\\text{.}\\text{6 m}[\/latex], [latex]\\text{65}\\text{.}\\text{1\u00ba}[\/latex] north of east<\/p>\n<p id=\"import-auto-id1165296374871\">(b) [latex]\\text{26}\\text{.}\\text{6 m}[\/latex], [latex]\\text{65}\\text{.}\\text{1\u00ba}[\/latex] south of west<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296576869\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296576870\">\n<p id=\"import-auto-id1165296261580\">Show that the <em data-effect=\"italics\"><em data-effect=\"italics\">order<\/em><\/em> of addition of three vectors does not affect their sum. Show this property by choosing any three vectors [latex]\\mathbf{A}[\/latex], [latex]\\mathbf{B}[\/latex], and [latex]\\mathbf{C}[\/latex], all having different lengths and directions. Find the sum [latex]\\text{A&nbsp;+&nbsp;B&nbsp;+&nbsp;C}[\/latex] then find their sum when added in a different order and show the result is the same. (There are five other orders in which [latex]\\mathbf{A}[\/latex], [latex]\\mathbf{B}[\/latex], and [latex]\\mathbf{C}[\/latex] can be added; choose only one.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298800490\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298879927\">\n<p id=\"import-auto-id1165296261582\">Show that the sum of the vectors discussed in <a href=\"#fs-id1165296679497\" class=\"autogenerated-content\">(Figure)<\/a> gives the result shown in <a href=\"#import-auto-id1165296298190\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298695430\">\n<p id=\"import-auto-id1165298828433\">[latex]\\text{52}\\text{.}\\text{9 m}[\/latex], [latex]\\text{90}\\text{.}\\text{1\u00ba}[\/latex] with respect to the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298704732\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298994968\">\n<p id=\"import-auto-id1165296335201\">Find the magnitudes of velocities [latex]{v}_{\\text{A}}[\/latex] and [latex]{v}_{\\text{B}}[\/latex] in <a href=\"#import-auto-id1165296217666\" class=\"autogenerated-content\">(Figure)<\/a><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296217666\">\n<div class=\"bc-figcaption figcaption\">The two velocities [latex]{\\mathbf{\\text{v}}}_{\\text{A}}[\/latex] and [latex]{\\mathbf{\\text{v}}}_{\\text{B}}[\/latex] add to give a total [latex]{\\mathbf{\\text{v}}}_{\\text{tot}}[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296217667\" data-alt=\"On the graph velocity vector V sub A begins at the origin and is inclined to x axis at an angle of twenty two point five degrees. From the head of vector V sub A another vector V sub B begins. The resultant of the two vectors, labeled V sub tot, is inclined to vector V sub A at twenty six point five degrees and to the vector V sub B at twenty three point zero degrees. V sub tot has a magnitude of 6.72 meters per second.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_23a.jpg\" data-media-type=\"image\/wmf\" alt=\"On the graph velocity vector V sub A begins at the origin and is inclined to x axis at an angle of twenty two point five degrees. From the head of vector V sub A another vector V sub B begins. The resultant of the two vectors, labeled V sub tot, is inclined to vector V sub A at twenty six point five degrees and to the vector V sub B at twenty three point zero degrees. V sub tot has a magnitude of 6.72 meters per second.\" width=\"250\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298735190\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296330739\">\n<p id=\"import-auto-id1165296227138\">Find the components of [latex]{v}_{\\text{tot}}[\/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 in <a href=\"#import-auto-id1165296217666\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298808711\">\n<p id=\"import-auto-id1165296217502\"><em data-effect=\"italics\">x<\/em>-component 4.41 m\/s<\/p>\n<p id=\"import-auto-id1165298695433\"><em data-effect=\"italics\">y<\/em>-component 5.07 m\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296227129\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296227130\">\n<p id=\"import-auto-id1165298978795\">Find the components of [latex]{v}_{\\text{tot}}[\/latex] along a set of perpendicular axes rotated [latex]\\text{30\u00ba}[\/latex] counterclockwise relative to those in <a href=\"#import-auto-id1165296217666\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\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-id1165296580417\">\n<dt>component (of a 2-d vector)<\/dt>\n<dd id=\"fs-id1165296249638\">a piece of a vector that points in either the vertical or the horizontal direction; every 2-d vector can be expressed as a sum of two vertical and horizontal vector components<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298785711\">\n<dt>commutative<\/dt>\n<dd id=\"fs-id1165296218975\">refers to the interchangeability of order in a function; vector addition is commutative because the order in which vectors are added together does not affect the final sum<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298785715\">\n<dt>direction (of a vector)<\/dt>\n<dd id=\"fs-id1165296376119\">the orientation of a vector in space<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298785717\">\n<dt>head (of a vector)<\/dt>\n<dd id=\"fs-id1165298657414\">the end point of a vector; the location of the tip of the vector\u2019s arrowhead; also referred to as the \u201ctip\u201d<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298837859\">\n<dt>head-to-tail method<\/dt>\n<dd id=\"fs-id1165298948455\">a method of adding vectors in which the tail of each vector is placed at the head of the previous vector<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298837863\">\n<dt>magnitude (of a vector)<\/dt>\n<dd id=\"fs-id1165298543790\">the length or size of a vector; magnitude is a scalar quantity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298837865\">\n<dt>resultant<\/dt>\n<dd id=\"fs-id1165298458853\"> the sum of two or more vectors<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298863819\">\n<dt>resultant vector<\/dt>\n<dd id=\"fs-id1165298621908\">the vector sum of two or more vectors<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298863821\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id1165298797862\">a quantity with magnitude but no direction<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298863823\">\n<dt>tail<\/dt>\n<dd id=\"fs-id1165298761576\">the start point of a vector; opposite to the head or tip of the arrow<\/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, subtraction, and multiplication.<\/li>\n<li>Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296227310\">\n<div class=\"bc-figcaption figcaption\">Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai\u2019i to Moloka\u2019i has a number of legs, or journey segments. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. (credit: US Geological Survey)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298679996\" data-alt=\"Some Hawaiian Islands like Kauai Oahu, Molokai, Lanai, Maui, Kahoolawe, and Hawaii are shown. On the scale map of Hawaiian Islands the path of a journey is shown moving from Hawaii to Molokai. The path of the journey is turning at different angles and finally reaching its destination. The displacement of the journey is shown with the help of a straight line connecting its starting point and the destination.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_00a.jpg\" data-media-type=\"image\/png\" alt=\"Some Hawaiian Islands like Kauai Oahu, Molokai, Lanai, Maui, Kahoolawe, and Hawaii are shown. On the scale map of Hawaiian Islands the path of a journey is shown moving from Hawaii to Molokai. The path of the journey is turning at different angles and finally reaching its destination. The displacement of the journey is shown with the help of a straight line connecting its starting point and the destination.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165296240221\">\n<h1 data-type=\"title\">Vectors in Two Dimensions<\/h1>\n<p id=\"import-auto-id1165298552138\">A <span data-type=\"term\" id=\"import-auto-id1165296389850\">vector<\/span> is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector\u2019s magnitude and pointing in the direction of the vector.<\/p>\n<p id=\"import-auto-id1165298918938\"><a href=\"#import-auto-id1165298666909\" class=\"autogenerated-content\">(Figure)<\/a> shows such a <em data-effect=\"italics\">graphical representation of a vector<\/em>, using as an example the total displacement for the person walking in a city considered in <a href=\"\/contents\/21d0e217-d50f-4901-af75-905e738eb4c4@4\">Kinematics in Two Dimensions: An Introduction<\/a>. We shall use the notation that a boldface symbol, such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b8bafaa394c07803fd77e1225e6cf6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>, stands for a vector. Its magnitude is represented by the symbol in italics, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/>, and its direction by <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;\" \/>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165296218458\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Vectors in this Text<\/div>\n<p>In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-094a27d245a761fc4d5c43863730d6b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/>, which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, and the direction of the variable will be given by an 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;\" \/>.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298666909\">\n<div class=\"bc-figcaption figcaption\">A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2aa6007afab3d3c2c721266ea80fddd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#49;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> north of east.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298713446\" data-alt=\"A graph is shown. On the axes the scale is set to one block is equal to one unit. A helicopter starts moving from the origin at an angle of twenty nine point one degrees above the x axis. The current position of the helicopter is ten point three blocks along its line of motion. The destination of the helicopter is the point which is nine blocks in the positive x direction and five blocks in the positive y direction. The positive direction of the x axis is east and the positive direction of the y axis is north.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_01.jpg\" data-media-type=\"image\/jpg\" alt=\"A graph is shown. On the axes the scale is set to one block is equal to one unit. A helicopter starts moving from the origin at an angle of twenty nine point one degrees above the x axis. The current position of the helicopter is ten point three blocks along its line of motion. The destination of the helicopter is the point which is nine blocks in the positive x direction and five blocks in the positive y direction. The positive direction of the x axis is east and the positive direction of the y axis is north.\" width=\"325\" \/><\/span><\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298918248\">\n<div class=\"bc-figcaption figcaption\">To describe the resultant vector for the person walking in a city considered in <a href=\"#import-auto-id1165298666909\" class=\"autogenerated-content\">(Figure)<\/a> graphically, draw an arrow to represent the total displacement vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b8bafaa394c07803fd77e1225e6cf6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. Using a protractor, draw a line at an 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;\" \/> relative to the east-west axis. The length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> of the arrow is proportional to the vector\u2019s magnitude and is measured along the line with a ruler. In this example, the magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> of the vector is 10.3 units, 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;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-be0c163fe793ebfdfdb7df551b901fae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#57;&#46;&#49;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> north of east.\n<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296263736\" data-alt=\"On a graph a vector is shown. It is inclined at an angle theta equal to twenty nine point one degrees above the positive x axis. A protractor is shown to the right of the x axis to measure the angle. A ruler is also shown parallel to the vector to measure its length. The ruler shows that the length of the vector is ten point three units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"On a graph a vector is shown. It is inclined at an angle theta equal to twenty nine point one degrees above the positive x axis. A protractor is shown to the right of the x axis to measure the angle. A ruler is also shown parallel to the vector to measure its length. The ruler shows that the length of the vector is ten point three units.\" height=\"250\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298995028\">\n<h1 data-type=\"title\">Vector Addition: Head-to-Tail Method<\/h1>\n<p id=\"import-auto-id1165298553339\">The <span data-type=\"term\" id=\"import-auto-id1165298552505\">head-to-tail method<\/span> is a graphical way to add vectors, described in <a href=\"#import-auto-id1165298643218\" class=\"autogenerated-content\">(Figure)<\/a> below and in the steps following. The <span data-type=\"term\">tail<\/span> of the vector is the starting point of the vector, and the <span data-type=\"term\" id=\"import-auto-id1165298982372\">head<\/span> (or tip) of a vector is the final, pointed end of the arrow.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298643218\">\n<div class=\"bc-figcaption figcaption\"><strong>Head-to-Tail Method:<\/strong> The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in <a href=\"#import-auto-id1165298666909\" class=\"autogenerated-content\">(Figure)<\/a>.  (a) Draw a vector representing the displacement to the east. (b) Draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first, east-pointing vector. (c) Draw a line from the tail of the east-pointing vector to the head of the north-pointing vector to form the sum or <span data-type=\"term\">resultant vector<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b8bafaa394c07803fd77e1225e6cf6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. The length of the arrow <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b8bafaa394c07803fd77e1225e6cf6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> is proportional to the vector\u2019s magnitude and is measured to be 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) <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;\" \/> is measured with a protractor to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a352c402c6c92f62682f1ed5d0180afc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298533716\" data-alt=\"In part a, a vector of magnitude of nine units and making an angle of theta is equal to zero degrees is drawn from the origin and along the positive direction of x axis. In part b a vector of magnitude of nine units and making an angle of theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical arrow from the head of the horizontal arrow is drawn. In part c a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vectorD.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_03.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a, a vector of magnitude of nine units and making an angle of theta is equal to zero degrees is drawn from the origin and along the positive direction of x axis. In part b a vector of magnitude of nine units and making an angle of theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical arrow from the head of the horizontal arrow is drawn. In part c a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vectorD.\" width=\"500\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165296543683\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 1.<\/em><\/strong><em data-effect=\"italics\"><em data-effect=\"italics\">Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor<\/em><\/em>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298876451\"><span data-type=\"media\" id=\"import-auto-id1165298800021\" data-alt=\"In part a, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis.\" height=\"200\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298805929\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 2.<\/em><\/strong> Now draw an arrow to represent the second vector (5 blocks to the north). <em data-effect=\"italics\">Place the tail of the second vector at the head of the first vector<\/em>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298818267\"><span data-type=\"media\" id=\"import-auto-id1165298672083\" data-alt=\"In part b, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical vector from the head of the horizontal vector is drawn.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part b, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical vector from the head of the horizontal vector is drawn.\" height=\"200\" \/><\/span><\/div>\n<p id=\"import-auto-id1165296690060\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 3.<\/em><\/strong><em data-effect=\"italics\">If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail<\/em>.<\/p>\n<p id=\"import-auto-id1165298943938\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 4.<\/em><\/strong><em data-effect=\"italics\">Draw an arrow from the tail of the first vector to the head of the last vector<\/em>. This is the <span data-type=\"term\" id=\"import-auto-id1165296311722\">resultant<\/span>, or the sum, of the other vectors.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165299000967\"><span data-type=\"media\" id=\"import-auto-id1165298799464\" data-alt=\"In part c, a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of the x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vector D.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part c, a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of the x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vector D.\" width=\"200\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298478340\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 5.<\/em><\/strong> To get the <span data-type=\"term\" id=\"import-auto-id1165298794109\">magnitude<\/span> of the resultant, <em data-effect=\"italics\">measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.)<\/em><\/p>\n<p id=\"import-auto-id1165298643122\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Step 6. <\/em><\/strong>To get the <span data-type=\"term\" id=\"import-auto-id1165298932041\">direction<\/span> of the resultant, <em data-effect=\"italics\"><em data-effect=\"italics\">measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.)<\/em><\/em><\/p>\n<p id=\"import-auto-id1165298452161\">The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296298332\">\n<div data-type=\"title\" class=\"title\">Adding Vectors Graphically Using the Head-to-Tail Method: A Woman Takes a Walk<\/div>\n<p id=\"import-auto-id1165298774683\">Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-58f89034a6498660970fa026d3d3a8b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> north of east. Then, she walks 23.0 m heading <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-456b987c72e1b778d082c7fc2009d9eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> north of east. Finally, she turns and walks 32.0 m in a direction 68.0\u00b0 south of east.<\/p>\n<p id=\"import-auto-id1165296336458\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1165298723286\">Represent each displacement vector graphically with an arrow, labeling the first <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2515aa44e115f2fe5bba4f79688ee56d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/>, the second <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92d94f2b1c3e25a4fdd5e7615e340179_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, and the third <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;\" \/>, making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted <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<p id=\"import-auto-id1165298842510\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165298879632\">(1) Draw the three displacement vectors.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296232338\"><span data-type=\"media\" id=\"import-auto-id1165298941346\" data-alt=\"On the graph a vector of magnitude twenty three meters and inclined above the x axis at an angle theta-b equal to fifteen degrees is shown. This vector is labeled as B.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_08.jpg\" data-media-type=\"image\/jpg\" alt=\"On the graph a vector of magnitude twenty three meters and inclined above the x axis at an angle theta-b equal to fifteen degrees is shown. This vector is labeled as B.\" width=\"500\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298699961\">(2) Place the vectors head to tail retaining both their initial magnitude and direction.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298788198\"><span data-type=\"media\" id=\"import-auto-id1165296306377\" data-alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_09.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis.\" width=\"250\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298517765\">(3) Draw the resultant vector, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-527329d796318b47489b7e327e81cf9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298786300\"><span data-type=\"media\" id=\"import-auto-id1165298835331\" data-alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_10.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector.\" width=\"250\" \/><\/span><\/div>\n<p id=\"import-auto-id1165296319738\">(4) Use a ruler to measure the magnitude 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;\" \/>, and a protractor to measure the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-527329d796318b47489b7e327e81cf9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298931707\"><span data-type=\"media\" id=\"import-auto-id1165296287125\" data-alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty meter and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector. A ruler is placed along the vector R to measure it. Also there is a protractor to measure the angle.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_11a.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty meter and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector. A ruler is placed along the vector R to measure it. Also there is a protractor to measure the angle.\" width=\"220\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298598693\">In this case, the total displacement <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;\" \/> is seen to have a magnitude of 50.0 m and to lie in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-952e159b1c176860803f1b67243d3b58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> south of east. By using its magnitude and direction, this vector can be expressed as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-94f30cde010972fb6d751d46b982ea2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#46;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"90\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fa1628a31ca360e1f9ae5d132496f65b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"55\" style=\"vertical-align: 0px;\" \/> south of east.<\/p>\n<p id=\"import-auto-id1165298639141\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id1165296455316\">      The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in <a href=\"#import-auto-id1165298931858\" class=\"autogenerated-content\">(Figure)<\/a> and we will still get the same solution.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298931858\"><span data-type=\"media\" id=\"import-auto-id1165296377152\" data-alt=\"In this figure a vector C with a negative slope is drawn from the origin. Then from the head of the vector C another vector A with positive slope is drawn and then another vector B with negative slope from the head of the vector A is drawn. From the tail of the vector C a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector B. The vector R is known as the resultant vector.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_12.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a vector C with a negative slope is drawn from the origin. Then from the head of the vector C another vector A with positive slope is drawn and then another vector B with negative slope from the head of the vector A is drawn. From the tail of the vector C a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector B. The vector R is known as the resultant vector.\" width=\"275\" \/><\/span><\/div>\n<p id=\"import-auto-id1165296613647\">Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is <span data-type=\"term\">commutative<\/span>. Vectors can be added in any order.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-686a19ee47658adde86ea4d18481374b_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;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"123\" style=\"vertical-align: -2px;\" \/><\/div>\n<p id=\"import-auto-id1165298670064\">(This is true for the addition of ordinary numbers as well\u2014you get the same result whether you add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ea2752646fe366f79505a20a2feacb6_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;&#50;&#125;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><br \/>\n or<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a7247d4c802f5de70c4311fae1577e05_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;&#51;&#125;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -2px;\" \/>, for example).<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298779158\">\n<h1 data-type=\"title\">Vector Subtraction<\/h1>\n<p id=\"import-auto-id1165298786530\">Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> from <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/p>\n<p>, written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c0d3db9d7798f502df3c13ec4f7ffee_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;&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/p>\n<p>, we must first define what we mean by subtraction. The <em data-effect=\"italics\">negative<\/em> of a vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<p>is defined to be <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;\" \/>; that is, graphically <em data-effect=\"italics\">the negative of any vector has the same magnitude but the opposite direction<\/em>, as shown in <a href=\"#import-auto-id1165298692950\" class=\"autogenerated-content\">(Figure)<\/a>. In other words, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> has the same length as <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;\" \/>, but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298692950\">\n<div class=\"bc-figcaption figcaption\">The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is the negative 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;\" \/>; it has the same length but opposite direction.      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296266911\" data-alt=\"Two vectors are shown. One of the vectors is labeled as vector   in north east direction. The other vector is of the same magnitude and is in the opposite direction to that of vector B. This vector is denoted as negative B.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_13a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two vectors are shown. One of the vectors is labeled as vector   in north east direction. The other vector is of the same magnitude and is in the opposite direction to that of vector B. This vector is denoted as negative B.\" height=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298788891\">The <em data-effect=\"italics\"><em data-effect=\"italics\">subtraction<\/em><\/em> of vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> from vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/> is then simply defined to be the addition 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;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/>. Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-454\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7bb596b846c5a98f2b6e73ea2636aa07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#32;&#45;&#32;&#66;&#32;&#61;&#32;&#65;&#32;&#43;&#32;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#66;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1165298555304\">This is analogous to the subtraction of scalars (where, for example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-67495452e1de845137250614aeab6177_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#32;&#45;&#32;&#50;&#32;&#61;&#32;&#53;&#32;&#43;&#32;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/>). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296679497\">\n<div data-type=\"title\" class=\"title\">Subtracting Vectors Graphically: A Woman Sailing a Boat<\/div>\n<p id=\"import-auto-id1165298586222\">A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7089a17f1cb9b88ba54b8b4d6a0288db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> north of east from her current location, and then travel 30.0 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a58b4af663a96aef2d11bcb1fb912782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#50;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -1px;\" \/> north of east (or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ed39369e2718cce795f47da3c95033a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> west of north). If the woman makes a mistake and travels in the <em data-effect=\"italics\"><em data-effect=\"italics\">opposite<\/em><\/em> direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296408744\"><span data-type=\"media\" id=\"import-auto-id1165296232464\" data-alt=\"A vector of magnitude twenty seven point five meters is shown. It is inclined to the horizontal at an angle of sixty six degrees. Another vector of magnitude thirty point zero meters is shown. It is inclined to the horizontal at an angle of one hundred and twelve degrees.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_14.jpg\" data-media-type=\"image\/jpg\" alt=\"A vector of magnitude twenty seven point five meters is shown. It is inclined to the horizontal at an angle of sixty six degrees. Another vector of magnitude thirty point zero meters is shown. It is inclined to the horizontal at an angle of one hundred and twelve degrees.\" width=\"450\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298644722\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1165298832023\">We can represent the first leg of the trip with a vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/>, and the second leg of the trip with a vector <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>. The dock is located at a location <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5ed5d87591570139bf3571ff7a2ead3_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;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"47\" style=\"vertical-align: -2px;\" \/>. If the woman mistakenly travels in the <em data-effect=\"italics\">opposite<\/em> direction for the second leg of the journey, she will travel a distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>  (30.0 m) in the direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-302d39cc4777df1725fbb996a0fef8df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&ordm;&#45;&#49;&#49;&#50;&ordm;&#61;&#54;&#56;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"115\" style=\"vertical-align: -1px;\" \/>  south of east. We represent this as <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;\" \/>, as shown below. The vector <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;\" \/> has the same magnitude as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> but is in the opposite direction. Thus, she will end up at a location <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d3fd61467ee3da3009220655f4d60af_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;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#66;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/>, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c0d3db9d7798f502df3c13ec4f7ffee_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;&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/>. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296408745\"><span data-type=\"media\" id=\"import-auto-id1165296232465\" data-alt=\"A vector labeled negative B is inclined at an angle of sixty-eight degrees below a horizontal line. A dotted line in the reverse direction inclined at one hundred and twelve degrees above the horizontal line is also shown.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_15a.jpg\" data-media-type=\"image\/jpg\" alt=\"A vector labeled negative B is inclined at an angle of sixty-eight degrees below a horizontal line. A dotted line in the reverse direction inclined at one hundred and twelve degrees above the horizontal line is also shown.\" width=\"200\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298473863\">We will perform vector addition to compare the location of the dock, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-751fdf97e2d61c30f9be1105fc1ba0cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#32;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#43;&#32;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"53\" style=\"vertical-align: -2px;\" \/>, with the location at which the woman mistakenly arrives, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b63232d47b391cbae9b867378f02c3c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#32;&#43;&#32;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#66;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"import-auto-id1165298943436\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296259545\">(1) To determine the location at which the woman arrives by accident, draw vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/> and <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;\" \/>.<\/p>\n<p id=\"import-auto-id1165298618212\">(2) Place the vectors head to tail.<\/p>\n<p id=\"import-auto-id1165296239091\">(3) Draw the resultant vector <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<p id=\"import-auto-id1165298732001\">(4) Use a ruler and protractor to measure 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;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298476927\"><span data-type=\"media\" id=\"import-auto-id1165298786656\" data-alt=\"Vectors A and negative B are connected in head to tail method. Vector A is inclined with horizontal with positive slope and vector negative B with a negative slope. The resultant of these two vectors is shown as a vector R from tail of A to the head of negative B. The length of the resultant is twenty three point zero meters and has a negative slope of seven point five degrees.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_16a.jpg\" data-media-type=\"image\/jpg\" alt=\"Vectors A and negative B are connected in head to tail method. Vector A is inclined with horizontal with positive slope and vector negative B with a negative slope. The resultant of these two vectors is shown as a vector R from tail of A to the head of negative B. The length of the resultant is twenty three point zero meters and has a negative slope of seven point five degrees.\" width=\"300\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298710088\">In this case, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3c9ada70d8d542c36615f575b2eedcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#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=\"12\" width=\"90\" style=\"vertical-align: 0px;\" \/><br \/>\n    and<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1eebd5e45be72bcd6414aa899e7b5222_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: 0px;\" \/><br \/>\n    south of east.<\/p>\n<p id=\"import-auto-id1165298996239\">(5) To determine the location of the dock, we repeat this method to add vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>. We obtain the resultant vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2b818b70369758d2e4a72e1890b0ae4_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;&#116;&#101;&#120;&#116;&#123;&#39;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296298190\"><span data-type=\"media\" id=\"import-auto-id1165296574919\" data-alt=\"A vector A inclined at sixty six degrees with horizontal is shown. From the head of this vector another vector B is started. Vector B is inclined at one hundred and twelve degrees with the horizontal. Another vector labeled as R prime from the tail of vector A to the head of vector B is drawn. The length of this vector is fifty two point nine meters and its inclination with the horizontal is shown as ninety point one degrees. Vector R prime is equal to the sum of vectors A and B.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_17a.jpg\" data-media-type=\"image\/jpg\" alt=\"A vector A inclined at sixty six degrees with horizontal is shown. From the head of this vector another vector B is started. Vector B is inclined at one hundred and twelve degrees with the horizontal. Another vector labeled as R prime from the tail of vector A to the head of vector B is drawn. The length of this vector is fifty two point nine meters and its inclination with the horizontal is shown as ninety point one degrees. Vector R prime is equal to the sum of vectors A and B.\" width=\"250\" \/><\/span><\/div>\n<p id=\"import-auto-id1165298652611\">In this case <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-180f9d50b0a4f46e8e4ee93fd8b5514b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#61;&#32;&#53;&#50;&#46;&#57;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"92\" style=\"vertical-align: 0px;\" \/><br \/>\n    and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f3197a7c3fe9a1451fc39f6eb3175174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#46;&#49;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"63\" style=\"vertical-align: -1px;\" \/><br \/>\n    &nbsp;north&nbsp;of&nbsp;east.<\/p>\n<p id=\"import-auto-id1165296220926\">We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.<\/p>\n<p id=\"import-auto-id1165298719642\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id1165296245111\">Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298652611\">\n<h1 data-type=\"title\">Multiplication of Vectors and Scalars<\/h1>\n<p id=\"import-auto-id1165298868552\">If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fa44d3723c2bb315cb6a7fd140d2788c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#125;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#50;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"73\" style=\"vertical-align: 0px;\" \/>, or 82.5 m, in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cd72e93e8c0f11cfcb210e32ea6cb363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#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. This is an example of multiplying a vector by a positive <span data-type=\"term\" id=\"import-auto-id1165296219603\">scalar<\/span>. Notice that the magnitude changes, but the direction stays the same.<\/p>\n<p id=\"import-auto-id1165298838383\">If the scalar is negative, then multiplying a vector by it changes the vector\u2019s magnitude and gives the new vector the <em data-effect=\"italics\"><em data-effect=\"italics\">opposite<\/em><\/em> direction. For example, if you multiply by \u20132, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When 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 multiplied by a scalar <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>,<\/p>\n<ul id=\"fs-id1165298531076\">\n<li id=\"import-auto-id1165298651742\">the magnitude of the vector becomes the absolute value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>[latex]A[\/latex],<\/li>\n<li id=\"import-auto-id1165298881521\">if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is positive, the direction of the vector does not change,<\/li>\n<li id=\"import-auto-id1165298455414\">if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is negative, the direction is reversed.<\/li>\n<\/ul>\n<p id=\"import-auto-id1165298960255\">In our case, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3dd8938ccec5cab45beecb5268738eaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b1c7cede3e65bbd298207cbd927a9d06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#50;&#55;&#46;&#53;&#32;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"84\" style=\"vertical-align: 0px;\" \/>. Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1\/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298819725\">\n<h1 data-type=\"title\">Resolving a Vector into Components<\/h1>\n<p id=\"import-auto-id1165298553346\">In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular <span data-type=\"term\" id=\"import-auto-id1165298555736\">components <\/span>of a single vector, for example the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211;<em data-effect=\"italics\"> and<\/em> <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-components, or the north-south and east-west components.<\/p>\n<p id=\"import-auto-id1165298883158\">For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ca243dd53c7eff266a640e9ac32d10c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> north of east and want to find out how many blocks east and north had to be walked. This method is called <em data-effect=\"italics\"><em data-effect=\"italics\">finding the components (or parts)<\/em><\/em> of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in <a href=\"\/contents\/69062f44-56d2-4111-88ff-f599727c4ed1@12\">Projectile Motion<\/a>, and much more when we cover <strong>forces<\/strong>  in <a href=\"\/contents\/02f52a02-2484-4ccb-bbd4-3c94edaa8e09@4\">Dynamics: Newton\u2019s Laws of Motion<\/a>. Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in <a href=\"\/contents\/b9739bfd-dc9d-4f0a-b037-dc22884d30f3@10\">Vector Addition and Subtraction: Analytical Methods<\/a> are ideal for finding vector components.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Maze Game<\/div>\n<p id=\"eip-id2002304\">Learn about position, velocity, and acceleration in the &#8220;Arena of Pain&#8221;. Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1434453\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/589973fcfa456bcabda2138e415688acf6b73e6f\/maze-game_en.jar\">Maze Game<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_3.2\" data-alt=\"\"><a href=\"\/resources\/589973fcfa456bcabda2138e415688acf6b73e6f\/maze-game_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-id1165298622440\">\n<h1 data-type=\"title\">Summary<\/h1>\n<ul id=\"fs-id1165298751188\">\n<li id=\"import-auto-id1165296253334\">The <strong>graphical method of adding vectors<\/strong> <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;\" \/> involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector\n<p><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 defined such that <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1eaa66669b37af81d8ee543aa7063032_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;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -2px;\" \/>. 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;\" \/> are then determined with a ruler and protractor, respectively.<\/p>\n<\/li>\n<li id=\"import-auto-id1165298573640\">The <strong>graphical method of subtracting vector <\/strong> <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;\" \/> involves adding the opposite of 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;\" \/>, which is defined as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7a2fef11baf14285d9757ed215596f8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>. In this case, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0fdf07dece40eb02c222be394c5f1666_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#66;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"175\" style=\"vertical-align: -4px;\" \/>. Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector <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;\" \/>.<\/li>\n<li id=\"import-auto-id1165296680072\">Addition of vectors is <span data-type=\"term\" id=\"import-auto-id1165296680069\">commutative<\/span> such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e0b32f294986474c804e9c07ec470093_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;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"118\" style=\"vertical-align: -2px;\" \/> .<\/li>\n<li id=\"import-auto-id1165296269519\">The <span data-type=\"term\" id=\"import-auto-id1165298982089\">head-to-tail method<\/span> of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.<\/li>\n<li id=\"import-auto-id1165298819524\">If 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;\" \/> is multiplied by a scalar quantity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, the magnitude of the product is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9411abdc6bb18064511f5e62068d2b6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is positive, the direction of the product points in the same direction as <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;\" \/>; if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is negative, the direction of the product points in the opposite direction as <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;\" \/>.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1165299003649\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298732767\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298730553\">\n<p id=\"import-auto-id1165298740934\">Which of the following is a vector: a person\u2019s height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this book, the Earth\u2019s population, the acceleration of gravity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298775836\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298806849\">\n<p id=\"import-auto-id1165298595710\">Give a specific example of a vector, stating its magnitude, units, and direction.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298783616\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298725379\">\n<p id=\"import-auto-id1165296264337\">What do vectors and scalars have in common? How do they differ?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296576944\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298861181\">\n<p id=\"import-auto-id1165296579324\">Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298840401\"><span data-type=\"media\" id=\"import-auto-id1165298840402\" data-alt=\"At the southwest corner of the figure is a cabin and in the northeast corner is a lake. A vector S with a length five point zero kilometers connects the cabin to the lake at an angle of 40 degrees north of east. Two winding paths labeled Path 1 and Path 2 represent the routes travelled from the cabin to the lake.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_18a.jpg\" data-media-type=\"image\/wmf\" alt=\"At the southwest corner of the figure is a cabin and in the northeast corner is a lake. A vector S with a length five point zero kilometers connects the cabin to the lake at an angle of 40 degrees north of east. Two winding paths labeled Path 1 and Path 2 represent the routes travelled from the cabin to the lake.\" width=\"275\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298981819\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298981820\">\n<p id=\"import-auto-id1165296218026\">If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in <a href=\"#import-auto-id1165296384452\" class=\"autogenerated-content\">(Figure)<\/a>. What other information would he need to get to Sacramento?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296384452\"><span data-type=\"media\" id=\"import-auto-id1165296384453\" 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.\" height=\"300\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298754450\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298754451\">\n<p id=\"import-auto-id1165298998383\">Suppose you take two steps <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>  (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5ed5d87591570139bf3571ff7a2ead3_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;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"47\" style=\"vertical-align: -2px;\" \/> the sum of the lengths of the two steps?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298761188\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298993266\">\n<p id=\"import-auto-id1165296716271\">Explain why it is not possible to add a scalar to a vector.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296242489\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298645243\">\n<p id=\"import-auto-id1165296408227\">If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1165298586130\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"import-auto-id1165298672665\"><strong data-effect=\"bold\">Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.<\/strong><\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298745593\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296363125\">\n<p id=\"import-auto-id1165298838476\">Find the following for path A in <a href=\"#import-auto-id1165298872310\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298872310\">\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-id1165298872312\" data-alt=\"A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meters each. The path of A extends to three blocks towards north and then one block towards east. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish.\"><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 meters each. The path of A extends to three blocks towards north and then one block towards east. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish.\" width=\"400\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296403819\">\n<p id=\"import-auto-id1165298835475\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc6e835be7fb42ae7326408067a4c364_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id1165296220703\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9382725f6e5965e9b121826c7d1afefd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#55;&#57;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-686e8e424d25fcef83b08eb8f4222854_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#46;&#52;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> east of north<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298474424\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298770529\">\n<p id=\"import-auto-id1165298723075\">Find the following for path B in <a href=\"#import-auto-id1165298872310\" class=\"autogenerated-content\">(Figure)<\/a>: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298867580\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296248676\">\n<p id=\"import-auto-id1165296365282\">Find the north and east components of the displacement for the hikers shown in <a href=\"#import-auto-id1165298840401\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296301840\">\n<p id=\"import-auto-id1165298650835\">north component 3.21 km, east component 3.83 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298536705\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296255759\">\n<p id=\"import-auto-id1165296243127\">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-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, as in <a href=\"#import-auto-id1165296241785\" 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-fc62f66e36989f135c7b1788068c5ad0_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;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -2px;\" \/>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296241785\">\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-id1165298744046\" data-alt=\"In this figure coordinate axes are shown. Vector A from the origin towards the negative of x axis is shown. From the head of the vector A another vector B is drawn towards the positive direction of y axis. The resultant R of these two vectors is shown as a vector from the tail of vector A to the head of vector B. This vector R is inclined at an angle theta with the negative x axis.\"><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 this figure coordinate axes are shown. Vector A from the origin towards the negative of x axis is shown. From the head of the vector A another vector B is drawn towards the positive direction of y axis. The resultant R of these two vectors is shown as a vector from the tail of vector A to the head of vector B. This vector R is inclined at an angle theta with the negative x axis.\" width=\"275\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298797729\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298797730\">\n<p id=\"import-auto-id1165298942186\">Suppose you first walk 12.0 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a35a9fd757241829645ed20e214d672c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> west of north and then 20.0 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ab2aeced1ef7bc58296d06bd266c34d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> south of west. 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-id1165296430663\" class=\"autogenerated-content\">(Figure)<\/a>, then this problem finds their sum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-682384262689060dd6b27825501bcdf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#32;&#61;&#32;&#65;&#32;&#43;&#32;&#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-id1165296430663\"><span data-type=\"media\" id=\"import-auto-id1165296430664\" data-alt=\"In the given figure coordinates axes are shown. Vector A with tail at origin is inclined at an angle of twenty degrees with the positive direction of x axis. The magnitude of vector A is twelve meters. Another vector B is starts from the head of vector A and inclined at an angle of forty degrees with the horizontal. The resultant R of the vectors A and B is also drawn from the tail of vector A to the head of vector B. The inclination of vector R is theta with the horizontal.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_22a.jpg\" data-media-type=\"image\/wmf\" alt=\"In the given figure coordinates axes are shown. Vector A with tail at origin is inclined at an angle of twenty degrees with the positive direction of x axis. The magnitude of vector A is twelve meters. Another vector B is starts from the head of vector A and inclined at an angle of forty degrees with the horizontal. The resultant R of the vectors A and B is also drawn from the tail of vector A to the head of vector B. The inclination of vector R is theta with the horizontal.\" width=\"250\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298995751\">\n<p id=\"import-auto-id1165298542917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3036655a5fe2e1a26f66bd91127206f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"51\" style=\"vertical-align: -1px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2403c5260fd5e07d5167e3aa70cead95_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;&#54;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> south of west<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298849088\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296252134\">\n<p id=\"import-auto-id1165298727195\">Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg <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;\" \/>, which is 20.0 m in a direction exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aba75ce0d9b36d567c38e44174eb2fb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/> south of west, and then leg <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;\" \/>, which is 12.0 m in a direction exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a35a9fd757241829645ed20e214d672c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> west of north. (This problem shows that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-debdf6db3cf9084feb69a6b37f33ea81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#66;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"126\" style=\"vertical-align: -2px;\" \/>.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298560552\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296243866\">\n<p id=\"import-auto-id1165298727198\">(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ab2aeced1ef7bc58296d06bd266c34d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> north of east (which is equivalent to subtracting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" 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, to finding <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f857f8fa776618185a942cdf5e09a21d_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;&#125;&#125;&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"89\" style=\"vertical-align: -1px;\" \/>). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ab2aeced1ef7bc58296d06bd266c34d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> south of west and then 12.0 m in a direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c918e80bba36ea1d2e565858d4a55186_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> east of south (which is equivalent to subtracting <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74cd9b5d36a15a85016531d432a0de7c_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: -1px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_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;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> \u2014that is, to finding <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a99a5bb8698b3804a568bb4a9c6dd6fc_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;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#61;&#45;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#125;&#125;&#92;&#112;&#114;&#105;&#109;&#101;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"149\" style=\"vertical-align: -1px;\" \/>). Show that this is the case.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296252966\">\n<p id=\"import-auto-id1165296541089\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5ccc288bb5c3fa0240b209acb3065172_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#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-867abeb14945e152e6b4908aca2c1d9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> north of east<\/p>\n<p id=\"import-auto-id1165296374871\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5ccc288bb5c3fa0240b209acb3065172_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#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-867abeb14945e152e6b4908aca2c1d9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> south of west<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296576869\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296576870\">\n<p id=\"import-auto-id1165296261580\">Show that the <em data-effect=\"italics\"><em data-effect=\"italics\">order<\/em><\/em> of addition of three vectors does not affect their sum. Show this property by choosing any 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-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;\" \/>, 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;\" \/>, all having different lengths and directions. Find the sum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-997aaaaef92baced60d48680c212efb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#32;&#43;&#32;&#66;&#32;&#43;&#32;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"89\" style=\"vertical-align: -2px;\" \/> then find their sum when added in a different order and show the result is the same. (There are five other orders in which <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-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;\" \/>, 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;\" \/> can be added; choose only one.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298800490\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298879927\">\n<p id=\"import-auto-id1165296261582\">Show that the sum of the vectors discussed in <a href=\"#fs-id1165296679497\" class=\"autogenerated-content\">(Figure)<\/a> gives the result shown in <a href=\"#import-auto-id1165296298190\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298695430\">\n<p id=\"import-auto-id1165298828433\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-409f7c1d0707d02e234cf5c996bec705_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-928833f93363480bf23cdc7232cbd2f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> with respect to the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298704732\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298994968\">\n<p id=\"import-auto-id1165296335201\">Find the magnitudes of velocities <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6a8a69db5ebd43a2b73ae01190561ff1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-deb3fd81cbd6640d21252212000fc9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\" \/> in <a href=\"#import-auto-id1165296217666\" class=\"autogenerated-content\">(Figure)<\/a><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296217666\">\n<div class=\"bc-figcaption figcaption\">The two velocities <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c725c763ac40fe9cc497ae70cb150528_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;&#118;&#125;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-374116c8373e7095eb4487b35258253c_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;&#118;&#125;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -3px;\" \/> add to give a total <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b51d7d1f6279b4cd8e87d48595cece9_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;&#118;&#125;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: -3px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296217667\" data-alt=\"On the graph velocity vector V sub A begins at the origin and is inclined to x axis at an angle of twenty two point five degrees. From the head of vector V sub A another vector V sub B begins. The resultant of the two vectors, labeled V sub tot, is inclined to vector V sub A at twenty six point five degrees and to the vector V sub B at twenty three point zero degrees. V sub tot has a magnitude of 6.72 meters per second.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_02_23a.jpg\" data-media-type=\"image\/wmf\" alt=\"On the graph velocity vector V sub A begins at the origin and is inclined to x axis at an angle of twenty two point five degrees. From the head of vector V sub A another vector V sub B begins. The resultant of the two vectors, labeled V sub tot, is inclined to vector V sub A at twenty six point five degrees and to the vector V sub B at twenty three point zero degrees. V sub tot has a magnitude of 6.72 meters per second.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298735190\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296330739\">\n<p id=\"import-auto-id1165296227138\">Find the components of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9d3f321a6dd11d6f45b30f64be347b0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\" \/> 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 in <a href=\"#import-auto-id1165296217666\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298808711\">\n<p id=\"import-auto-id1165296217502\"><em data-effect=\"italics\">x<\/em>-component 4.41 m\/s<\/p>\n<p id=\"import-auto-id1165298695433\"><em data-effect=\"italics\">y<\/em>-component 5.07 m\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296227129\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296227130\">\n<p id=\"import-auto-id1165298978795\">Find the components of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9d3f321a6dd11d6f45b30f64be347b0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\" \/> along a set of perpendicular axes rotated <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7e404b535ec15d8300b0b1148704da97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> counterclockwise relative to those in <a href=\"#import-auto-id1165296217666\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\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-id1165296580417\">\n<dt>component (of a 2-d vector)<\/dt>\n<dd id=\"fs-id1165296249638\">a piece of a vector that points in either the vertical or the horizontal direction; every 2-d vector can be expressed as a sum of two vertical and horizontal vector components<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298785711\">\n<dt>commutative<\/dt>\n<dd id=\"fs-id1165296218975\">refers to the interchangeability of order in a function; vector addition is commutative because the order in which vectors are added together does not affect the final sum<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298785715\">\n<dt>direction (of a vector)<\/dt>\n<dd id=\"fs-id1165296376119\">the orientation of a vector in space<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298785717\">\n<dt>head (of a vector)<\/dt>\n<dd id=\"fs-id1165298657414\">the end point of a vector; the location of the tip of the vector\u2019s arrowhead; also referred to as the \u201ctip\u201d<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298837859\">\n<dt>head-to-tail method<\/dt>\n<dd id=\"fs-id1165298948455\">a method of adding vectors in which the tail of each vector is placed at the head of the previous vector<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298837863\">\n<dt>magnitude (of a vector)<\/dt>\n<dd id=\"fs-id1165298543790\">the length or size of a vector; magnitude is a scalar quantity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298837865\">\n<dt>resultant<\/dt>\n<dd id=\"fs-id1165298458853\"> the sum of two or more vectors<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298863819\">\n<dt>resultant vector<\/dt>\n<dd id=\"fs-id1165298621908\">the vector sum of two or more vectors<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298863821\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id1165298797862\">a quantity with magnitude but no direction<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298863823\">\n<dt>tail<\/dt>\n<dd id=\"fs-id1165298761576\">the start point of a vector; opposite to the head or tip of the arrow<\/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-179","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\/179","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\/179\/revisions"}],"predecessor-version":[{"id":180,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/179\/revisions\/180"}],"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\/179\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=179"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=179"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=179"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}