{"id":269,"date":"2020-04-19T15:58:49","date_gmt":"2020-04-19T19:58:49","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/chapter\/3-2-vector-addition-and-subtraction-graphical-methods-2\/"},"modified":"2020-07-04T21:15:59","modified_gmt":"2020-07-05T01:15:59","slug":"3-2-vector-addition-and-subtraction-graphical-methods-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/chapter\/3-2-vector-addition-and-subtraction-graphical-methods-2\/","title":{"raw":"4.2 Vector Addition and Subtraction","rendered":"4.2 Vector Addition and Subtraction"},"content":{"raw":"<div>\r\n<div class=\"bcc-box bcc-highlight\">\r\n<h3>Summary<\/h3>\r\n<div>\r\n<ul>\r\n \t<li>Define and apply the rules of vector addition and subtraction.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Vectors in One Dimension<\/h1>\r\nA <strong><span id=\"import-auto-id1165296389850\">vector<\/span> <\/strong>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 the horizontal axis, + corresponds to movement to the right and - corresponds to movement to the left. In the vertical axis, + corresponds to upward movement and - corresponds to downward movement.\r\n\r\nIf all of the vectors are acting along the horizontal axis (x), vectors can be added or subtracted together like regular numbers. If all of the vectors are acting along the vertical axis (y), vectors can be added or subtracted together like regular numbers. For example, if a person runs 8 m to the right stops and then runs 10 m to the right, their final displacement is (+8m +10m) 18m. If the same person then walked 13 m to the left, the final displacement would be: +18m - 13 m = +5m.\r\n\r\nIf one of the vectors is acting along the horizontal axis (x) and one is acting along the vertical axis (y), you will have to use a different strategy, suitable for adding or subtracting vectors in two dimensions (x and y).\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1.602em;font-weight: bold\">Vectors in Two Dimensions (One vector acting in y, one in x)<\/span>\r\n\r\n<section id=\"fs-id1165296240221\">\r\n<p id=\"import-auto-id1165298552138\">In two dimensions (2-d), 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>\r\n<p id=\"import-auto-id1165298918938\"><a class=\"autogenerated-content\" href=\"#import-auto-id1165298666909\">Figure 1 <\/a>shows such a <em>graphical representation of a vector<\/em>, using as an example the total displacement for the person running in a city considered in the previous section. The person ran 9 blocks east and 5 blocks north and we want to know his final displacement. We shall use the notation that a symbol with an arrow over it, such as [latex]\\vec{\\text{D}}[\/latex], stands for a vector. Its magnitude is represented by the symbol in italics<em>, <strong>D<\/strong>,<\/em> and its direction by <em><strong>\u03b8<\/strong><\/em>.<\/p>\r\n\r\n<div id=\"fs-id1165296218458\" class=\"note\">\r\n<div class=\"textbox shaded\">\r\n<div class=\"note\">\r\n<h3 class=\"title\">VECTORS IN THIS TEXT<\/h3>\r\nIn this text, we will represent a vector with an arrow over a symbol. For example, we will represent the quantity force with the vector [latex]\\vec{\\text{F}}[\/latex], which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as <em><strong>F<\/strong><\/em>, and the direction of the variable will be given by an angle <em><strong>\u03b8<\/strong><\/em>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<figure id=\"import-auto-id1165298666909\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_01-1-1.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\" height=\"324\" \/> <strong>Figure 1.<\/strong> A person runs 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle <strong>29.1<sup>o<\/sup><\/strong> north of east.[\/caption]<\/figure>\r\n<figure id=\"import-auto-id1165298918248\"><\/figure>\r\n<\/section><section id=\"fs-id1165298995028\">\r\n<h1>Vector Addition: Tip-to-Tail Method<\/h1>\r\n<p id=\"import-auto-id1165298553339\">The <strong>tip<span id=\"import-auto-id1165298552505\">-to-tail method<\/span><\/strong> is a graphical way to add two vectors, when one is vertical and the other horizontal. It is described in <a class=\"autogenerated-content\" href=\"#import-auto-id1165298643218\">Figure 2<\/a> below and in the steps following. The <strong>tail <\/strong>of the vector is the starting point of the vector, and the <b>tip<\/b>\u00a0of a vector is the final, pointed end of the arrow.<\/p>\r\n\r\n<figure id=\"import-auto-id1165298643218\"><figcaption><\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"500\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_03-1-1.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\" height=\"454\" \/> <strong>Figure 2.<\/strong> Tip-to-Tail Method: \u00a0(a) Draw a vector representing the displacement to the east. (b) At the tip of the first vector, draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first. (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 <strong>resultant vector D<\/strong>. The length of the arrow <strong>D<\/strong> is calculated using Pythagorean's theorem and measures 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) <strong>\u03b8<\/strong> is calculated withe SOH CAH TOA to be <strong>29.1<sup>0<\/sup><\/strong>.[\/caption]<\/figure>\r\n<strong><em>Step 1.<\/em><\/strong> <em><em>Draw an arrow to represent the first vector (9 blocks to the east) using a ruler<\/em><\/em>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"344\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_04a-1-1.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.\" width=\"344\" height=\"200\" \/> <strong>Figure 3.<\/strong>[\/caption]\r\n\r\n<strong><em>Step 2.<\/em><\/strong> Now draw an arrow to represent the second vector (5 blocks to the north). <em>Place the tail of the second vector at the tip of the first vector<\/em>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"343\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_05a-1-1.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.\" width=\"343\" height=\"200\" \/> <strong>Figure 4.<\/strong>[\/caption]\r\n<p id=\"import-auto-id1165296690060\"><strong><em>Step 3.<\/em><\/strong> <em>If there are more than two vectors, start by adding all the horizontal vectors together to have one resultant x vector and add all the vertical vectors together to have one resultant y vector. Use the two resultants in the tip-to-tail method.<\/em><\/p>\r\n<strong><em>Step 4.<\/em><\/strong> <em>Draw an arrow from the tail of the first vector to the head of the last vector<\/em>. This is the <strong><span id=\"import-auto-id1165296311722\">resultant<\/span><\/strong>, or the sum, of the other vectors.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"200\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_06a-1-1.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\" height=\"400\" \/> <strong>Figure 5.<\/strong>[\/caption]\r\n<p id=\"import-auto-id1165298478340\"><strong><em>Step 5.<\/em><\/strong> To get the <strong><span id=\"import-auto-id1165298794109\">magnitude<\/span> <\/strong>of the resultant<em>, we will use the Pythagorean theorem to determine the length of the hypotenuse of this newly formed right angle triangle. (D = \u221ax<sup>2<\/sup>+y2)<\/em><\/p>\r\n<p id=\"import-auto-id1165298643122\"><strong><em>Step 6. <\/em><\/strong>To get the <strong><span id=\"import-auto-id1165298932041\">direction<\/span> <\/strong>of the resultant, <em><em>we will use trigonometric relationships.\u00a0<\/em><\/em><\/p>\r\n<p style=\"text-align: center\">tan\u00a0\u03b8 = opposite\/adjacent<\/p>\r\n<p style=\"text-align: center\">tan\u00a0\u03b8 = y\/x<\/p>\r\n\r\n<\/section><section id=\"fs-id1165298819725\">\r\n<h1><span style=\"font-size: 1.602em;font-weight: bold\">Finding the vectors components<\/span><\/h1>\r\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 id=\"import-auto-id1165298555736\"><strong>components<\/strong> <\/span>of a single vector, for example the <em><em>x<\/em><\/em>-<em> and<\/em> <em><em>y<\/em><\/em>-components, or the north-south and east-west components.<\/p>\r\nFor example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction <strong>29.0\u00b0<\/strong> north of east and want to find out how many blocks east and north had to be walked. This method is called <em><em>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. You can draw the components (x and y) of the vector and the resultant (hypothenuse) as a right-angle triangle and use your knowledge of trigonometry to solve for the length of the components (x and y).\r\n\r\n<\/section><section id=\"fs-id1165298622440\" class=\"section-summary\">\r\n<h1>Vector in Two-Dimensions (Adding vectors acting at angles)<\/h1>\r\n<p id=\"import-auto-id1165298553339\">More realistically, you will have to add vectors that are not acting perfectly along the horizontal axis or the vertical axis. If you have to add vectors that are acting at angles and that are composed of a horizontal component and a vertical component. The preferred strategy in this case, would be to break the vectors down into their horizontal and vertical components. You can then add all of the horizontal components together and all the vertical components together to end up with two vectors acting perfectly along each axis. Use the tip-to-tail method to reconstruct your final results vector. Remember that your final answer should have a magnitude and direction (an angle in this case).<\/p>\r\n\r\n<h1>Summary<\/h1>\r\n<ul id=\"fs-id1165298751188\">\r\n \t<li id=\"import-auto-id1165296253334\">The<strong>\u00a0method of adding vectors <\/strong>[latex]\\vec{\\text{A}}[\/latex] and [latex]\\vec{\\text{B}}[\/latex] involves drawing vectors on a graph and adding them using the tip-to-tail method. The resultant vector [latex]\\vec{\\text{R}}[\/latex] is defined such that [latex]\\vec{\\text{A}}+\\vec{\\text{B}}=\\vec{\\text{R}}[\/latex]. The magnitude and direction of [latex]\\vec{\\text{R}}[\/latex] are then determined with the Pythagorean theorem and trigonometric ratios (SOH CAH TOA), respectively.<\/li>\r\n \t<li id=\"import-auto-id1165296269519\">The <strong><span id=\"import-auto-id1165298982089\">tip-to-tail method<\/span><\/strong> 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>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1165299003649\" class=\"conceptual-questions\">\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Conceptual Questions<\/h3>\r\n<div id=\"fs-id1165298732767\" class=\"exercise\">\r\n<div id=\"fs-id1165298730553\" class=\"problem\">\r\n<p id=\"import-auto-id1165298740934\"><strong>1: <\/strong>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>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165298775836\" class=\"exercise\">\r\n<div id=\"fs-id1165298806849\" class=\"problem\">\r\n<p id=\"import-auto-id1165298595710\"><strong>2: <\/strong>Give a specific example of a vector, stating its magnitude, units, and direction.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165298783616\" class=\"exercise\">\r\n<div id=\"fs-id1165298725379\" class=\"problem\">\r\n<p id=\"import-auto-id1165296264337\"><strong>3: <\/strong>What do vectors and scalars have in common? How do they differ?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165296576944\" class=\"exercise\">\r\n<div id=\"fs-id1165298861181\" class=\"problem\">\r\n<p id=\"import-auto-id1165296579324\"><strong>4: <\/strong>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>\r\n\r\n<figure id=\"import-auto-id1165298840401\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"275\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_18a-1-1.jpg\" 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\" height=\"216\" \/> <strong>Figure 6.<\/strong>[\/caption]<\/figure>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165298981819\" class=\"exercise\">\r\n<div id=\"fs-id1165298981820\" class=\"problem\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165298754450\" class=\"exercise\">\r\n<div id=\"fs-id1165298754451\" class=\"problem\">\r\n<p id=\"import-auto-id1165298998383\"><strong>5: <\/strong>Suppose you take two steps A and B (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 A+B the sum of the lengths of the two steps?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165298761188\" class=\"exercise\">\r\n<div id=\"fs-id1165298993266\" class=\"problem\">\r\n<p id=\"import-auto-id1165296716271\"><strong>6: <\/strong>Explain why it is not possible to add a scalar to a vector.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165296242489\" class=\"exercise\">\r\n<div id=\"fs-id1165298645243\" class=\"problem\">\r\n<p id=\"import-auto-id1165296408227\"><strong>7:\u00a0<\/strong>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>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165298586130\" class=\"problems-exercises\">\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Problems &amp; Exercises<\/h3>\r\n<p id=\"import-auto-id1165298672665\"><strong>Solve these problems.\u00a0<\/strong><\/p>\r\n\r\n<div id=\"fs-id1165298536705\" class=\"exercise\">\r\n<div id=\"fs-id1165296255759\" class=\"problem\">\r\n<p id=\"import-auto-id1165296243127\"><strong>1: <\/strong>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 $\\vec{\\text{A }}$ and $\\vec{\\text{B}}$, as in <a class=\"autogenerated-content\" href=\"#import-auto-id1165296241785\">Figure 7<\/a>, then this problem asks you to find their sum $\\vec{\\text{R}}=\\vec{\\text{A}}+\\vec{\\text{B}}$.)<\/p>\r\n\r\n<figure id=\"import-auto-id1165296241785\"><figcaption><\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"275\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_21a-2-1.jpg\" 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\" height=\"286\" \/> <strong>Figure 7.<\/strong> The two displacements <strong>A<\/strong> and <strong>B<\/strong> add to give a total displacement<strong> R<\/strong> having magnitude <em><strong>R<\/strong> <\/em>and direction <strong>\u03b8<\/strong>.[\/caption]<\/figure>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165298797729\" class=\"exercise\">\r\n<div id=\"fs-id1165298797730\" class=\"problem\">\r\n<p id=\"import-auto-id1165298942186\"><strong>2: <\/strong>Suppose you first walk 12.0 m in a direction 20\u00b0 west of north and then 20.0 m in a direction 40.0\u00b0 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?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"import-auto-id1165296580417\" class=\"definition\">\r\n \t<dt>component (of a 2-d vector)<\/dt>\r\n \t<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>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298785711\" class=\"definition\">\r\n \t<dt>commutative<\/dt>\r\n \t<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>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298785715\" class=\"definition\">\r\n \t<dt>direction (of a vector)<\/dt>\r\n \t<dd id=\"fs-id1165296376119\">the orientation of a vector in space<\/dd>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298785717\" class=\"definition\">\r\n \t<dt>tip (of a vector)<\/dt>\r\n \t<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>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298837859\" class=\"definition\">\r\n \t<dt>tip-to-tail method<\/dt>\r\n \t<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>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298837863\" class=\"definition\">\r\n \t<dt>magnitude (of a vector)<\/dt>\r\n \t<dd id=\"fs-id1165298543790\">the length or size of a vector; magnitude is a scalar quantity<\/dd>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298837865\" class=\"definition\">\r\n \t<dt>resultant<\/dt>\r\n \t<dd id=\"fs-id1165298458853\">the sum of two or more vectors<\/dd>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298863819\" class=\"definition\">\r\n \t<dt>resultant vector<\/dt>\r\n \t<dd id=\"fs-id1165298621908\">the vector sum of two or more vectors<\/dd>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298863821\" class=\"definition\">\r\n \t<dt>scalar<\/dt>\r\n \t<dd>a quantity with magnitude but no direction<\/dd>\r\n<\/dl>\r\n<dl id=\"import-auto-id1165298863823\" class=\"definition\">\r\n \t<dt>tail<\/dt>\r\n \t<dd id=\"fs-id1165298761576\">the start point of a vector; opposite to the head or tip of the arrow<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Solutions<\/h3>\r\n<strong>Problems &amp; Exercises<\/strong>\r\n\r\n<strong>2: <\/strong>[latex]\\boldsymbol{19.5\\textbf{ m}},\\boldsymbol{4.65^o}[\/latex] south of west\r\n\r\n<\/div>","rendered":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<ul>\n<li>Define and apply the rules of vector addition and subtraction.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<h1>Vectors in One Dimension<\/h1>\n<p>A <strong><span id=\"import-auto-id1165296389850\">vector<\/span> <\/strong>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 the horizontal axis, + corresponds to movement to the right and &#8211; corresponds to movement to the left. In the vertical axis, + corresponds to upward movement and &#8211; corresponds to downward movement.<\/p>\n<p>If all of the vectors are acting along the horizontal axis (x), vectors can be added or subtracted together like regular numbers. If all of the vectors are acting along the vertical axis (y), vectors can be added or subtracted together like regular numbers. For example, if a person runs 8 m to the right stops and then runs 10 m to the right, their final displacement is (+8m +10m) 18m. If the same person then walked 13 m to the left, the final displacement would be: +18m &#8211; 13 m = +5m.<\/p>\n<p>If one of the vectors is acting along the horizontal axis (x) and one is acting along the vertical axis (y), you will have to use a different strategy, suitable for adding or subtracting vectors in two dimensions (x and y).<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1.602em;font-weight: bold\">Vectors in Two Dimensions (One vector acting in y, one in x)<\/span><\/p>\n<section id=\"fs-id1165296240221\">\n<p id=\"import-auto-id1165298552138\">In two dimensions (2-d), 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 class=\"autogenerated-content\" href=\"#import-auto-id1165298666909\">Figure 1 <\/a>shows such a <em>graphical representation of a vector<\/em>, using as an example the total displacement for the person running in a city considered in the previous section. The person ran 9 blocks east and 5 blocks north and we want to know his final displacement. We shall use the notation that a symbol with an arrow over it, such as [latex]\\vec{\\text{D}}[\/latex], stands for a vector. Its magnitude is represented by the symbol in italics<em>, <strong>D<\/strong>,<\/em> and its direction by <em><strong>\u03b8<\/strong><\/em>.<\/p>\n<div id=\"fs-id1165296218458\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">VECTORS IN THIS TEXT<\/h3>\n<p>In this text, we will represent a vector with an arrow over a symbol. For example, we will represent the quantity force with the vector [latex]\\vec{\\text{F}}[\/latex], which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as <em><strong>F<\/strong><\/em>, and the direction of the variable will be given by an angle <em><strong>\u03b8<\/strong><\/em>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<figure id=\"import-auto-id1165298666909\">\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_01-1-1.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\" height=\"324\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> A person runs 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle <strong>29.1<sup>o<\/sup><\/strong> north of east.<\/figcaption><\/figure>\n<\/figure>\n<figure id=\"import-auto-id1165298918248\"><\/figure>\n<\/section>\n<section id=\"fs-id1165298995028\">\n<h1>Vector Addition: Tip-to-Tail Method<\/h1>\n<p id=\"import-auto-id1165298553339\">The <strong>tip<span id=\"import-auto-id1165298552505\">-to-tail method<\/span><\/strong> is a graphical way to add two vectors, when one is vertical and the other horizontal. It is described in <a class=\"autogenerated-content\" href=\"#import-auto-id1165298643218\">Figure 2<\/a> below and in the steps following. The <strong>tail <\/strong>of the vector is the starting point of the vector, and the <b>tip<\/b>\u00a0of a vector is the final, pointed end of the arrow.<\/p>\n<figure id=\"import-auto-id1165298643218\"><figcaption><\/figcaption><figure style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_03-1-1.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\" height=\"454\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong> Tip-to-Tail Method: \u00a0(a) Draw a vector representing the displacement to the east. (b) At the tip of the first vector, draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first. (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 <strong>resultant vector D<\/strong>. The length of the arrow <strong>D<\/strong> is calculated using Pythagorean&#8217;s theorem and measures 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) <strong>\u03b8<\/strong> is calculated withe SOH CAH TOA to be <strong>29.1<sup>0<\/sup><\/strong>.<\/figcaption><\/figure>\n<\/figure>\n<p><strong><em>Step 1.<\/em><\/strong> <em><em>Draw an arrow to represent the first vector (9 blocks to the east) using a ruler<\/em><\/em>.<\/p>\n<figure style=\"width: 344px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_04a-1-1.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.\" width=\"344\" height=\"200\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<p><strong><em>Step 2.<\/em><\/strong> Now draw an arrow to represent the second vector (5 blocks to the north). <em>Place the tail of the second vector at the tip of the first vector<\/em>.<\/p>\n<figure style=\"width: 343px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_05a-1-1.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.\" width=\"343\" height=\"200\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<p id=\"import-auto-id1165296690060\"><strong><em>Step 3.<\/em><\/strong> <em>If there are more than two vectors, start by adding all the horizontal vectors together to have one resultant x vector and add all the vertical vectors together to have one resultant y vector. Use the two resultants in the tip-to-tail method.<\/em><\/p>\n<p><strong><em>Step 4.<\/em><\/strong> <em>Draw an arrow from the tail of the first vector to the head of the last vector<\/em>. This is the <strong><span id=\"import-auto-id1165296311722\">resultant<\/span><\/strong>, or the sum, of the other vectors.<\/p>\n<figure style=\"width: 200px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_06a-1-1.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\" height=\"400\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<p id=\"import-auto-id1165298478340\"><strong><em>Step 5.<\/em><\/strong> To get the <strong><span id=\"import-auto-id1165298794109\">magnitude<\/span> <\/strong>of the resultant<em>, we will use the Pythagorean theorem to determine the length of the hypotenuse of this newly formed right angle triangle. (D = \u221ax<sup>2<\/sup>+y2)<\/em><\/p>\n<p id=\"import-auto-id1165298643122\"><strong><em>Step 6. <\/em><\/strong>To get the <strong><span id=\"import-auto-id1165298932041\">direction<\/span> <\/strong>of the resultant, <em><em>we will use trigonometric relationships.\u00a0<\/em><\/em><\/p>\n<p style=\"text-align: center\">tan\u00a0\u03b8 = opposite\/adjacent<\/p>\n<p style=\"text-align: center\">tan\u00a0\u03b8 = y\/x<\/p>\n<\/section>\n<section id=\"fs-id1165298819725\">\n<h1><span style=\"font-size: 1.602em;font-weight: bold\">Finding the vectors components<\/span><\/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 id=\"import-auto-id1165298555736\"><strong>components<\/strong> <\/span>of a single vector, for example the <em><em>x<\/em><\/em>&#8211;<em> and<\/em> <em><em>y<\/em><\/em>-components, or the north-south and east-west components.<\/p>\n<p>For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction <strong>29.0\u00b0<\/strong> north of east and want to find out how many blocks east and north had to be walked. This method is called <em><em>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. You can draw the components (x and y) of the vector and the resultant (hypothenuse) as a right-angle triangle and use your knowledge of trigonometry to solve for the length of the components (x and y).<\/p>\n<\/section>\n<section id=\"fs-id1165298622440\" class=\"section-summary\">\n<h1>Vector in Two-Dimensions (Adding vectors acting at angles)<\/h1>\n<p id=\"import-auto-id1165298553339\">More realistically, you will have to add vectors that are not acting perfectly along the horizontal axis or the vertical axis. If you have to add vectors that are acting at angles and that are composed of a horizontal component and a vertical component. The preferred strategy in this case, would be to break the vectors down into their horizontal and vertical components. You can then add all of the horizontal components together and all the vertical components together to end up with two vectors acting perfectly along each axis. Use the tip-to-tail method to reconstruct your final results vector. Remember that your final answer should have a magnitude and direction (an angle in this case).<\/p>\n<h1>Summary<\/h1>\n<ul id=\"fs-id1165298751188\">\n<li id=\"import-auto-id1165296253334\">The<strong>\u00a0method of adding vectors <\/strong>[latex]\\vec{\\text{A}}[\/latex] and [latex]\\vec{\\text{B}}[\/latex] involves drawing vectors on a graph and adding them using the tip-to-tail method. The resultant vector [latex]\\vec{\\text{R}}[\/latex] is defined such that [latex]\\vec{\\text{A}}+\\vec{\\text{B}}=\\vec{\\text{R}}[\/latex]. The magnitude and direction of [latex]\\vec{\\text{R}}[\/latex] are then determined with the Pythagorean theorem and trigonometric ratios (SOH CAH TOA), respectively.<\/li>\n<li id=\"import-auto-id1165296269519\">The <strong><span id=\"import-auto-id1165298982089\">tip-to-tail method<\/span><\/strong> 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<\/ul>\n<\/section>\n<section id=\"fs-id1165299003649\" class=\"conceptual-questions\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<div id=\"fs-id1165298732767\" class=\"exercise\">\n<div id=\"fs-id1165298730553\" class=\"problem\">\n<p id=\"import-auto-id1165298740934\"><strong>1: <\/strong>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 id=\"fs-id1165298775836\" class=\"exercise\">\n<div id=\"fs-id1165298806849\" class=\"problem\">\n<p id=\"import-auto-id1165298595710\"><strong>2: <\/strong>Give a specific example of a vector, stating its magnitude, units, and direction.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165298783616\" class=\"exercise\">\n<div id=\"fs-id1165298725379\" class=\"problem\">\n<p id=\"import-auto-id1165296264337\"><strong>3: <\/strong>What do vectors and scalars have in common? How do they differ?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165296576944\" class=\"exercise\">\n<div id=\"fs-id1165298861181\" class=\"problem\">\n<p id=\"import-auto-id1165296579324\"><strong>4: <\/strong>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<figure id=\"import-auto-id1165298840401\">\n<figure style=\"width: 275px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_18a-1-1.jpg\" 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\" height=\"216\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165298981819\" class=\"exercise\">\n<div id=\"fs-id1165298981820\" class=\"problem\"><\/div>\n<\/div>\n<div id=\"fs-id1165298754450\" class=\"exercise\">\n<div id=\"fs-id1165298754451\" class=\"problem\">\n<p id=\"import-auto-id1165298998383\"><strong>5: <\/strong>Suppose you take two steps A and B (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 A+B the sum of the lengths of the two steps?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165298761188\" class=\"exercise\">\n<div id=\"fs-id1165298993266\" class=\"problem\">\n<p id=\"import-auto-id1165296716271\"><strong>6: <\/strong>Explain why it is not possible to add a scalar to a vector.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165296242489\" class=\"exercise\">\n<div id=\"fs-id1165298645243\" class=\"problem\">\n<p id=\"import-auto-id1165296408227\"><strong>7:\u00a0<\/strong>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<\/section>\n<section id=\"fs-id1165298586130\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<p id=\"import-auto-id1165298672665\"><strong>Solve these problems.\u00a0<\/strong><\/p>\n<div id=\"fs-id1165298536705\" class=\"exercise\">\n<div id=\"fs-id1165296255759\" class=\"problem\">\n<p id=\"import-auto-id1165296243127\"><strong>1: <\/strong>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 $\\vec{\\text{A }}$ and $\\vec{\\text{B}}$, as in <a class=\"autogenerated-content\" href=\"#import-auto-id1165296241785\">Figure 7<\/a>, then this problem asks you to find their sum $\\vec{\\text{R}}=\\vec{\\text{A}}+\\vec{\\text{B}}$.)<\/p>\n<figure id=\"import-auto-id1165296241785\"><figcaption><\/figcaption><figure style=\"width: 275px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-content\/uploads\/sites\/972\/2020\/04\/Figure_03_02_21a-2-1.jpg\" 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\" height=\"286\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong> The two displacements <strong>A<\/strong> and <strong>B<\/strong> add to give a total displacement<strong> R<\/strong> having magnitude <em><strong>R<\/strong> <\/em>and direction <strong>\u03b8<\/strong>.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165298797729\" class=\"exercise\">\n<div id=\"fs-id1165298797730\" class=\"problem\">\n<p id=\"import-auto-id1165298942186\"><strong>2: <\/strong>Suppose you first walk 12.0 m in a direction 20\u00b0 west of north and then 20.0 m in a direction 40.0\u00b0 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?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"import-auto-id1165296580417\" class=\"definition\">\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 id=\"import-auto-id1165298785711\" class=\"definition\">\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 id=\"import-auto-id1165298785715\" class=\"definition\">\n<dt>direction (of a vector)<\/dt>\n<dd id=\"fs-id1165296376119\">the orientation of a vector in space<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1165298785717\" class=\"definition\">\n<dt>tip (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 id=\"import-auto-id1165298837859\" class=\"definition\">\n<dt>tip-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 id=\"import-auto-id1165298837863\" class=\"definition\">\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 id=\"import-auto-id1165298837865\" class=\"definition\">\n<dt>resultant<\/dt>\n<dd id=\"fs-id1165298458853\">the sum of two or more vectors<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1165298863819\" class=\"definition\">\n<dt>resultant vector<\/dt>\n<dd id=\"fs-id1165298621908\">the vector sum of two or more vectors<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1165298863821\" class=\"definition\">\n<dt>scalar<\/dt>\n<dd>a quantity with magnitude but no direction<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1165298863823\" class=\"definition\">\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<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><strong>2: <\/strong>[latex]\\boldsymbol{19.5\\textbf{ m}},\\boldsymbol{4.65^o}[\/latex] south of west<\/p>\n<\/div>\n","protected":false},"author":71,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-269","chapter","type-chapter","status-publish","hentry"],"part":236,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/pressbooks\/v2\/chapters\/269","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/wp\/v2\/users\/71"}],"version-history":[{"count":10,"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/pressbooks\/v2\/chapters\/269\/revisions"}],"predecessor-version":[{"id":938,"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/pressbooks\/v2\/chapters\/269\/revisions\/938"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/pressbooks\/v2\/parts\/236"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/pressbooks\/v2\/chapters\/269\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/wp\/v2\/media?parent=269"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/pressbooks\/v2\/chapter-type?post=269"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/wp\/v2\/contributor?post=269"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/wp-json\/wp\/v2\/license?post=269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}