{"id":156,"date":"2021-03-01T08:51:24","date_gmt":"2021-03-01T13:51:24","guid":{"rendered":"http:\/\/pressbooks.library.upei.ca\/statics\/?post_type=chapter&#038;p=156"},"modified":"2025-07-28T09:33:56","modified_gmt":"2025-07-28T13:33:56","slug":"vector-form-notation","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.upei.ca\/statics\/chapter\/vector-form-notation\/","title":{"raw":"1.3 Vectors","rendered":"1.3 Vectors"},"content":{"raw":"<h1>1.3.1 Vector Components<\/h1>\r\nSome fun facts about vectors:\r\n<ul>\r\n \t<li>The vector is denoted with a line on top or bottom: [latex]\\vec A[\/latex] or <strong><span style=\"text-decoration: underline\">A<\/span><\/strong>.<\/li>\r\n \t<li>There are two parts of a vector ([latex]\\vec A[\/latex]): magnitude (A or |<span style=\"text-decoration: underline\">A<\/span>|) and direction ([latex]\\underline{\\hat{a}} [\/latex]): [latex]\\vec A = |\\underline{A}|*\\underline{\\hat{a}} [\/latex]<\/li>\r\n \t<li>In 2-dimensions, there are two <em>components: <\/em>x and y. In 3D, there are three components: x, y, and z.<\/li>\r\n \t<li>Vectors can be denoted using Cartesian form or brackets: [latex]\\vec A=A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}+A_z\\underline{\\hat{k}}[\/latex] or using the bracket form horizontally: [latex]\\vec A=[ A_x, A_y, A_z [\/latex] ] or vertically:\u00a0 [latex]\\vec A=\\begin{bmatrix}A_x\\\\A_y,\\\\A_z \\end{bmatrix}[\/latex]<\/li>\r\n \t<li>The magnitude (A or |<span style=\"text-decoration: underline\">A<\/span>|) is calculated using the Pythagorean theorem for each component in 2d: [latex] A = \\sqrt{{A}_{x}^{2}+{A}_{y}^{2}}[\/latex] and 3d: [latex]A = \\sqrt{{A}_{x}^{2}+{A}_{y}^{2}+{A}_{z}^{2}}[\/latex]<\/li>\r\n \t<li>The unit vector ([latex]\\underline{\\hat{u}}[\/latex]) represents the direction in cartesian form [latex]\\underline{\\hat{u}}=\\underline{\\hat{i}}+\\underline{\\hat{j}}+\\underline{\\hat{k}}[\/latex] or using bracket form: [ [latex]\\underline{\\hat{i}}, \\underline{\\hat{j}}, \\underline{\\hat{k}} [\/latex] ].<\/li>\r\n \t<li>The magnitude of the unit vector is 1 (denoted by the 'hat' on top) and it is unit-less: [latex]|\\underline{\\hat{u}} |= \\sqrt{{\\underline{\\hat{i}}}^{2}+{\\underline{\\hat{j}}}^{2}+{\\underline{\\hat{k}}}^{2}} = 1[\/latex]<\/li>\r\n \t<li>The unit vector can be calculated from the magnitude and vector: [latex]\\underline{\\hat{a}} =\\vec A\/|A|[\/latex]<\/li>\r\n<\/ul>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n\r\nIn 2d &amp; 3d:\r\n\r\n&nbsp;\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2952\/2018\/01\/31183906\/CNX_UPhysics_02_02_comp03.jpg\" alt=\"Vector A has horizontal x component A sub x equal to magnitude A sub x I hat and vertical y component A sub y equal to magnitude A sub y j hat. Vector A and the components form a right triangle with sides length magnitude A sub x and magnitude A sub y and hypotenuse magnitude A equal to the square root of A sub x squared plus A sub y squared. The angle between the horizontal side A sub x and the hypotenuse A is theta sub A.\" class=\"alignleft\" width=\"322\" height=\"232\" \/><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2952\/2018\/01\/31183920\/CNX_UPhysics_02_02_vector3D.jpg\" alt=\"Vector A in the x y z coordinate system extends from the origin. Vector A equals the sum of vectors A sub x, A sub y and A sub z. Vector A sub x is the x component along the x axis and has length A sub x I hat. Vector A sub y is the y component along the y axis and has length A sub y j hat. Vector A sub z is the z component along the z axis and has length A sub x k hat. The components form the sides of a rectangular box with sides length A sub x, A sub y, and A sub z.\" class=\"aligncenter\" width=\"289\" height=\"266\" style=\"padding-left: 0px;text-align: justify\" \/>\r\n\r\nSource: University Physics Volume 1, OpenStax CNX, <a href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-2-coordinate-systems-and-components-of-a-vector\/\">https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-2-coordinate-systems-and-components-of-a-vector\/<\/a>\r\n\r\n<\/div>\r\n<h1>1.3.2 Componentizing a Vector<\/h1>\r\n[caption id=\"attachment_714\" align=\"alignright\" width=\"200\"]<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-12.27.17-AM-300x293.png\" alt=\"A 2D coordinate plane with x and y axes. A point P is marked, connected to the origin by a line labeled r, forming an angle \u03b8 with the x-axis. The angle \u03b8 is shown with a dashed arc.\" class=\"wp-image-714\" width=\"200\" height=\"195\" \/> Source: Introductory Physics, Ryan Martin et al., <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a> p814[\/caption]\r\n\r\nIn 2d:\r\n\r\nTo find the components of a vector (<span style=\"text-decoration: underline\">A<\/span>) in 2 dimensions (the x and y portions A<sub>x<\/sub> and A<sub>y<\/sub>), use SOH CAH TOA:\r\n\r\n&nbsp;\r\n<div class=\"page\" title=\"Page 830\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<span style=\"text-decoration: underline\"><\/span>[latex]\\vec A=A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}[\/latex]\r\n\r\n<span>A<sub>x <\/sub><\/span><span>= |<span style=\"text-decoration: underline\">A<\/span>|<\/span><span> <\/span><span>cos(\u0398<\/span><span><\/span><span>)<\/span>\r\n\r\n<span>A<sub>y<\/sub> = |<span style=\"text-decoration: underline\">A<\/span>|<\/span><span> <\/span><span>sin(\u0398<\/span><span><\/span><span>) <\/span>\r\n\r\n<span style=\"text-decoration: underline\">|A|<\/span><sup>2<\/sup> = A<sub>x<\/sub><sup>2<\/sup> + A<sub>y<\/sub><sup>2\u00a0 <\/sup>(magnitude)\r\n\r\ntan(<span>\u0398<\/span>) = A<sub>y<\/sub> \/ A<sub>x<\/sub>\u00a0 \u00a0 (direction)\r\n\r\n<span style=\"text-decoration: underline\"><\/span><span>\r\n<\/span>\r\n\r\nIn 3d:\r\n\r\n[latex]\\vec A=A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}+A_z\\underline{\\hat{k}}[\/latex]\r\n\r\n<span>|<span style=\"text-decoration: underline\">A<\/span>|<\/span><sup>2<\/sup> = A<sub>x<\/sub><sup>2<\/sup> + A<sub>y<\/sub><sup>2<\/sup>+ A<sub>z<\/sub><sup>2 <\/sup>\u00a0 (magnitude)\r\n\r\n[latex]\r\n\\begin{aligned}\r\n&amp;\\hat{a}=\\frac{\\vec A}{|\\vec A|}\r\n\\end{aligned}=\\frac{{A}_{x} \\underline{\\hat{\\imath}}+A_{y} \\underline{\\hat{\\jmath}}+{A}_{z} \\underline{\\hat{k}}}{\\sqrt{\\left({A}_{x}\\right)^{2}+\\left({A}_{y}\\right)^{2}+\\left({A}_{z}\\right)^{2}}}\r\n[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>1.3.3 Position Vector<\/h1>\r\nThe position vector describes the position of an object or person from a predefined origin (a starting point, absolute 0, or some other point), for example the point where <span style=\"text-decoration: underline\">A<\/span> in the above image is point at. <span style=\"text-decoration: underline\">A<\/span> is the position vector. You can add individual position vectors to find the total position traveled (<span style=\"text-decoration: underline\">c<\/span> = <span style=\"text-decoration: underline\">a<\/span> + <span style=\"text-decoration: underline\">b<\/span>), for example if someone walks from one point on campus to another, they would rarely walk in one straight line like c. In the image below, imagine that there is a building in the square near where <span style=\"text-decoration: underline\">a<\/span> and <span style=\"text-decoration: underline\">b<\/span> meet, so the person couldn't take <span style=\"text-decoration: underline\">c<\/span> but had to walk around. The total distance traveled is |<span style=\"text-decoration: underline\">a|<\/span> + <span style=\"text-decoration: underline\">|b|<\/span>, not |<span style=\"text-decoration: underline\">c<\/span>| (because |<span style=\"text-decoration: underline\">c<\/span>| \u2260 |<span><span style=\"text-decoration: underline\">a<\/span>|<\/span> + <span>|<span style=\"text-decoration: underline\">b<\/span>|)<\/span>.\r\n<div class=\"textbox\">\r\n<div class=\"page\" title=\"Page 837\">\r\n\r\n[caption id=\"attachment_2664\" align=\"aligncenter\" width=\"300\"]<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-15-151515-1-300x186.png\" alt=\"A diagram showing vector addition in 2D: vectors a and b form a parallelogram, resulting in vector c. Equations for vector components are shown: c\u2093 = a\u2093 + b\u2093 and c\u1d67 = a\u1d67 + b\u1d67.\" width=\"300\" height=\"186\" class=\"wp-image-2664 size-medium\" \/> Source: Introductory Physics, Ryan Martin et al., <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a> page 821[\/caption]\r\n\r\nGeometric addition of the vectors a and b by placing them \u201chead to tail.\u201d\r\n\r\n<\/div>\r\n<div title=\"Page 837\">\r\n<div class=\"page\" title=\"Page 837\">\r\n<div class=\"page\" title=\"Page 837\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Page 837\">\r\n<div class=\"page\" title=\"Page 837\">\r\n<div class=\"page\" title=\"Page 837\">\r\n\r\nSubtraction works the same way, but instead of going from tail to head of the arrow, the reverse direction is taken, from head to tail. For example, <span style=\"text-decoration: underline\">a<\/span> = c-b<span style=\"text-decoration: underline\">,<\/span> follow c from tail to head, then go in the reverse direction of b from head to tail, and you end up at <span style=\"text-decoration: underline\">a.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>1.3.4 Vector Math<\/h1>\r\nHere's more official language to describe vectors:\r\n<div class=\"textbox\">\r\n\r\nVectors can be added together and multiplied by scalars. Vector addition is associative and commutative, and vector multiplication by a sum of scalars is distributive. Also, scalar multiplication by a sum of vectors is distributive:\r\n<p style=\"text-align: center\">[latex]\\alpha(\\vec A+\\vec B)=\\alpha\\vec A +\\alpha\\vec B[\/latex]<\/p>\r\nIn this equation, \u03b1 is any number (a scalar). For example, a vector antiparallel to vector [latex]\\vec A=A_x\\hat{i}+A_y\\hat{j}+A_z\\hat{k}[\/latex] can be expressed simply by multiplying [latex]\\vec A [\/latex] by the scalar \u03b1=-1:\r\n<p style=\"text-align: center\">[latex]-\\vec A=-A_x\\hat{i}-A_y\\hat{j}-A_z\\hat{k}[\/latex]<\/p>\r\n<p id=\"fs-id1167132504881\">The generalization of the number zero to vector algebra is called the\u00a0null vector, denoted by [latex]\\vec 0 [\/latex]. All components of the null vector are zero [latex] \\vec 0 = 0 \\hat{i} + 0 \\hat{j} + 0 \\hat{k}[\/latex] , so the null vector has no length and no direction.<\/p>\r\n<p id=\"fs-id1167133485200\">Two vectors [latex]\\vec A [\/latex]\u00a0and [latex]\\vec B [\/latex]\u00a0are\u00a0equal vectors\u00a0if and only if their difference is the null vector:<\/p>\r\n[latex]\\vec{0} = \\vec{A} - \\vec{B} = (A_x\\hat{i} + A_y\\hat{j} + A_z\\hat{k}) - (B_x\\hat{i} + B_y\\hat{j} + B_z\\hat{k}) \\\\ = (A_x - B_x)\\hat{i} + (A_y - B_y)\\hat{j} + (A_z - B_z)\\hat{k}[\/latex]\r\n<p style=\"text-align: center\"><span>[latex]\\space=(A_x - B_x)\\underline{\\hat{i}} + (A_y - B_y)\\underline{\\hat{j}} + (A_z - B_z)\\underline{\\hat{k}}[\/latex]\u00a0<\/span><\/p>\r\n<span>This vector equation means we must have simultaneously [latex]A_x-B_x=0[\/latex], [latex]A_y-B_y=0[\/latex], and [latex]A_z-B_z=0[\/latex]<\/span><span>. Hence, we can write [latex]\\vec A=\\vec B[\/latex] <\/span><span>if and only if the corresponding components of vectors\u00a0[latex]\\vec A[\/latex]<\/span><span> and [latex]\\vec B[\/latex]<\/span><span> are equal:<\/span>\r\n<p style=\"text-align: center\">[latex]\\vec A =\\vec B[\/latex]\u00a0\u00a0 if \u00a0 [latex]\\begin{bmatrix}A_x=B_x\\\\A_y=B_y\\\\A_z=B_z\\end{bmatrix}[\/latex]<\/p>\r\n<p id=\"fs-id1167133466190\">Two vectors are equal when their corresponding scalar components are equal.<\/p>\r\n<p id=\"fs-id1167133321055\">Resolving vectors into their scalar components (i.e., finding their scalar components) and expressing them analytically in vector component form allows us to use vector algebra to find sums or differences of many vectors <em>analytically <\/em>(i.e., using graphical methods). For example, to find the resultant of two vectors [latex]\\overset{\\to }{A}[\/latex]<span>\u00a0<\/span>and<span> [latex]\\vec B[\/latex]<\/span>, we simply add them component by component, as follows:<\/p>\r\n[latex]\\vec R=\\vec A + \\vec B=(A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}+A_z\\underline{\\hat{k}})+(B_x\\underline{\\hat{i}}+B_y\\underline{\\hat{j}}+B_z\\underline{\\hat{k}})=(A_x+B_x)\\underline{\\hat{i}}+(A_y+B_y)\\underline{\\hat{j}}+(A_z+B_z)\\underline{\\hat{k}}[\/latex]\r\n<p style=\"text-align: left\"><span>In this way,<\/span><span> scalar components of the resultant vector: [latex]\\vec R=(R_x\\underline{\\hat{i}}+R_y\\underline{\\hat{j}}+R_z\\underline{\\hat{k}})[\/latex].<\/span><\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{matrix}R_x = A_x+B_x\\\\R_y = A_y+B_y\\\\R_z = A_z+B_z\\end{matrix}[\/latex]<\/p>\r\n\r\n<div><\/div>\r\n<div id=\"fs-id1167132536774\" class=\"unnumbered\">Source: University Physics Volume 1, OpenStax CNX, <a href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-3-algebra-of-vectors\/\">https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-3-algebra-of-vectors\/<\/a><\/div>\r\n<\/div>\r\n<div><img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-1024x585.png\" alt=\"A campus map with labeled buildings and a yellow path from the FSDE building to the Library, showing vectors R\u2081 to R\u2087, each marked with red points and arrows. X and Y axes are shown near a soccer field.\" class=\"aligncenter wp-image-862 size-large\" width=\"1024\" height=\"585\" \/><\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<em>Basically: <\/em>Vectors help describe position, forces, and quantities. Vectors use components, magnitude, and direction (unit vector) to do so.\r\n\r\n<em>Application: <\/em>A hammock hangs at an angle from the wall. When a person is in the hammock, they are pulling on the wall with a force at an angle. This force vector could be componentized into x and y, using the angle and the weight of the person to calculate it.\r\n\r\n<em>Looking ahead: <\/em>The next place vectors will appear is in Moments in 1.6.\r\n\r\n<\/div>\r\n<\/div>","rendered":"<h1>1.3.1 Vector Components<\/h1>\n<p>Some fun facts about vectors:<\/p>\n<ul>\n<li>The vector is denoted with a line on top or bottom: [latex]\\vec A[\/latex] or <strong><span style=\"text-decoration: underline\">A<\/span><\/strong>.<\/li>\n<li>There are two parts of a vector ([latex]\\vec A[\/latex]): magnitude (A or |<span style=\"text-decoration: underline\">A<\/span>|) and direction ([latex]\\underline{\\hat{a}}[\/latex]): [latex]\\vec A = |\\underline{A}|*\\underline{\\hat{a}}[\/latex]<\/li>\n<li>In 2-dimensions, there are two <em>components: <\/em>x and y. In 3D, there are three components: x, y, and z.<\/li>\n<li>Vectors can be denoted using Cartesian form or brackets: [latex]\\vec A=A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}+A_z\\underline{\\hat{k}}[\/latex] or using the bracket form horizontally: [latex]\\vec A=[ A_x, A_y, A_z[\/latex] ] or vertically:\u00a0 [latex]\\vec A=\\begin{bmatrix}A_x\\\\A_y,\\\\A_z \\end{bmatrix}[\/latex]<\/li>\n<li>The magnitude (A or |<span style=\"text-decoration: underline\">A<\/span>|) is calculated using the Pythagorean theorem for each component in 2d: [latex]A = \\sqrt{{A}_{x}^{2}+{A}_{y}^{2}}[\/latex] and 3d: [latex]A = \\sqrt{{A}_{x}^{2}+{A}_{y}^{2}+{A}_{z}^{2}}[\/latex]<\/li>\n<li>The unit vector ([latex]\\underline{\\hat{u}}[\/latex]) represents the direction in cartesian form [latex]\\underline{\\hat{u}}=\\underline{\\hat{i}}+\\underline{\\hat{j}}+\\underline{\\hat{k}}[\/latex] or using bracket form: [ [latex]\\underline{\\hat{i}}, \\underline{\\hat{j}}, \\underline{\\hat{k}}[\/latex] ].<\/li>\n<li>The magnitude of the unit vector is 1 (denoted by the &#8216;hat&#8217; on top) and it is unit-less: [latex]|\\underline{\\hat{u}} |= \\sqrt{{\\underline{\\hat{i}}}^{2}+{\\underline{\\hat{j}}}^{2}+{\\underline{\\hat{k}}}^{2}} = 1[\/latex]<\/li>\n<li>The unit vector can be calculated from the magnitude and vector: [latex]\\underline{\\hat{a}} =\\vec A\/|A|[\/latex]<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<p>In 2d &amp; 3d:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2952\/2018\/01\/31183906\/CNX_UPhysics_02_02_comp03.jpg\" alt=\"Vector A has horizontal x component A sub x equal to magnitude A sub x I hat and vertical y component A sub y equal to magnitude A sub y j hat. Vector A and the components form a right triangle with sides length magnitude A sub x and magnitude A sub y and hypotenuse magnitude A equal to the square root of A sub x squared plus A sub y squared. The angle between the horizontal side A sub x and the hypotenuse A is theta sub A.\" class=\"alignleft\" width=\"322\" height=\"232\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2952\/2018\/01\/31183920\/CNX_UPhysics_02_02_vector3D.jpg\" alt=\"Vector A in the x y z coordinate system extends from the origin. Vector A equals the sum of vectors A sub x, A sub y and A sub z. Vector A sub x is the x component along the x axis and has length A sub x I hat. Vector A sub y is the y component along the y axis and has length A sub y j hat. Vector A sub z is the z component along the z axis and has length A sub x k hat. The components form the sides of a rectangular box with sides length A sub x, A sub y, and A sub z.\" class=\"aligncenter\" width=\"289\" height=\"266\" style=\"padding-left: 0px;text-align: justify\" \/><\/p>\n<p>Source: University Physics Volume 1, OpenStax CNX, <a href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-2-coordinate-systems-and-components-of-a-vector\/\">https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-2-coordinate-systems-and-components-of-a-vector\/<\/a><\/p>\n<\/div>\n<h1>1.3.2 Componentizing a Vector<\/h1>\n<figure id=\"attachment_714\" aria-describedby=\"caption-attachment-714\" style=\"width: 200px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-12.27.17-AM-300x293.png\" alt=\"A 2D coordinate plane with x and y axes. A point P is marked, connected to the origin by a line labeled r, forming an angle \u03b8 with the x-axis. The angle \u03b8 is shown with a dashed arc.\" class=\"wp-image-714\" width=\"200\" height=\"195\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-12.27.17-AM-300x293.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-12.27.17-AM-65x63.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-12.27.17-AM-225x220.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-12.27.17-AM-350x342.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-12.27.17-AM.png 580w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/><figcaption id=\"caption-attachment-714\" class=\"wp-caption-text\">Source: Introductory Physics, Ryan Martin et al., <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a> p814<\/figcaption><\/figure>\n<p>In 2d:<\/p>\n<p>To find the components of a vector (<span style=\"text-decoration: underline\">A<\/span>) in 2 dimensions (the x and y portions A<sub>x<\/sub> and A<sub>y<\/sub>), use SOH CAH TOA:<\/p>\n<p>&nbsp;<\/p>\n<div class=\"page\" title=\"Page 830\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><span style=\"text-decoration: underline\"><\/span>[latex]\\vec A=A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}[\/latex]<\/p>\n<p><span>A<sub>x <\/sub><\/span><span>= |<span style=\"text-decoration: underline\">A<\/span>|<\/span><span> <\/span><span>cos(\u0398<\/span><span><\/span><span>)<\/span><\/p>\n<p><span>A<sub>y<\/sub> = |<span style=\"text-decoration: underline\">A<\/span>|<\/span><span> <\/span><span>sin(\u0398<\/span><span><\/span><span>) <\/span><\/p>\n<p><span style=\"text-decoration: underline\">|A|<\/span><sup>2<\/sup> = A<sub>x<\/sub><sup>2<\/sup> + A<sub>y<\/sub><sup>2\u00a0 <\/sup>(magnitude)<\/p>\n<p>tan(<span>\u0398<\/span>) = A<sub>y<\/sub> \/ A<sub>x<\/sub>\u00a0 \u00a0 (direction)<\/p>\n<p><span style=\"text-decoration: underline\"><\/span><span><br \/>\n<\/span><\/p>\n<p>In 3d:<\/p>\n<p>[latex]\\vec A=A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}+A_z\\underline{\\hat{k}}[\/latex]<\/p>\n<p><span>|<span style=\"text-decoration: underline\">A<\/span>|<\/span><sup>2<\/sup> = A<sub>x<\/sub><sup>2<\/sup> + A<sub>y<\/sub><sup>2<\/sup>+ A<sub>z<\/sub><sup>2 <\/sup>\u00a0 (magnitude)<\/p>\n<p>[latex]\\begin{aligned}  &\\hat{a}=\\frac{\\vec A}{|\\vec A|}  \\end{aligned}=\\frac{{A}_{x} \\underline{\\hat{\\imath}}+A_{y} \\underline{\\hat{\\jmath}}+{A}_{z} \\underline{\\hat{k}}}{\\sqrt{\\left({A}_{x}\\right)^{2}+\\left({A}_{y}\\right)^{2}+\\left({A}_{z}\\right)^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h1>1.3.3 Position Vector<\/h1>\n<p>The position vector describes the position of an object or person from a predefined origin (a starting point, absolute 0, or some other point), for example the point where <span style=\"text-decoration: underline\">A<\/span> in the above image is point at. <span style=\"text-decoration: underline\">A<\/span> is the position vector. You can add individual position vectors to find the total position traveled (<span style=\"text-decoration: underline\">c<\/span> = <span style=\"text-decoration: underline\">a<\/span> + <span style=\"text-decoration: underline\">b<\/span>), for example if someone walks from one point on campus to another, they would rarely walk in one straight line like c. In the image below, imagine that there is a building in the square near where <span style=\"text-decoration: underline\">a<\/span> and <span style=\"text-decoration: underline\">b<\/span> meet, so the person couldn&#8217;t take <span style=\"text-decoration: underline\">c<\/span> but had to walk around. The total distance traveled is |<span style=\"text-decoration: underline\">a|<\/span> + <span style=\"text-decoration: underline\">|b|<\/span>, not |<span style=\"text-decoration: underline\">c<\/span>| (because |<span style=\"text-decoration: underline\">c<\/span>| \u2260 |<span><span style=\"text-decoration: underline\">a<\/span>|<\/span> + <span>|<span style=\"text-decoration: underline\">b<\/span>|)<\/span>.<\/p>\n<div class=\"textbox\">\n<div class=\"page\" title=\"Page 837\">\n<figure id=\"attachment_2664\" aria-describedby=\"caption-attachment-2664\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-15-151515-1-300x186.png\" alt=\"A diagram showing vector addition in 2D: vectors a and b form a parallelogram, resulting in vector c. Equations for vector components are shown: c\u2093 = a\u2093 + b\u2093 and c\u1d67 = a\u1d67 + b\u1d67.\" width=\"300\" height=\"186\" class=\"wp-image-2664 size-medium\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-15-151515-1-300x186.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-15-151515-1-65x40.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-15-151515-1-225x139.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-15-151515-1-350x217.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-15-151515-1.png 525w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-2664\" class=\"wp-caption-text\">Source: Introductory Physics, Ryan Martin et al., <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a> page 821<\/figcaption><\/figure>\n<p>Geometric addition of the vectors a and b by placing them \u201chead to tail.\u201d<\/p>\n<\/div>\n<div title=\"Page 837\">\n<div class=\"page\" title=\"Page 837\">\n<div class=\"page\" title=\"Page 837\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Page 837\">\n<div class=\"page\" title=\"Page 837\">\n<div class=\"page\" title=\"Page 837\">\n<p>Subtraction works the same way, but instead of going from tail to head of the arrow, the reverse direction is taken, from head to tail. For example, <span style=\"text-decoration: underline\">a<\/span> = c-b<span style=\"text-decoration: underline\">,<\/span> follow c from tail to head, then go in the reverse direction of b from head to tail, and you end up at <span style=\"text-decoration: underline\">a.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h1>1.3.4 Vector Math<\/h1>\n<p>Here&#8217;s more official language to describe vectors:<\/p>\n<div class=\"textbox\">\n<p>Vectors can be added together and multiplied by scalars. Vector addition is associative and commutative, and vector multiplication by a sum of scalars is distributive. Also, scalar multiplication by a sum of vectors is distributive:<\/p>\n<p style=\"text-align: center\">[latex]\\alpha(\\vec A+\\vec B)=\\alpha\\vec A +\\alpha\\vec B[\/latex]<\/p>\n<p>In this equation, \u03b1 is any number (a scalar). For example, a vector antiparallel to vector [latex]\\vec A=A_x\\hat{i}+A_y\\hat{j}+A_z\\hat{k}[\/latex] can be expressed simply by multiplying [latex]\\vec A[\/latex] by the scalar \u03b1=-1:<\/p>\n<p style=\"text-align: center\">[latex]-\\vec A=-A_x\\hat{i}-A_y\\hat{j}-A_z\\hat{k}[\/latex]<\/p>\n<p id=\"fs-id1167132504881\">The generalization of the number zero to vector algebra is called the\u00a0null vector, denoted by [latex]\\vec 0[\/latex]. All components of the null vector are zero [latex]\\vec 0 = 0 \\hat{i} + 0 \\hat{j} + 0 \\hat{k}[\/latex] , so the null vector has no length and no direction.<\/p>\n<p id=\"fs-id1167133485200\">Two vectors [latex]\\vec A[\/latex]\u00a0and [latex]\\vec B[\/latex]\u00a0are\u00a0equal vectors\u00a0if and only if their difference is the null vector:<\/p>\n<p>[latex]\\vec{0} = \\vec{A} - \\vec{B} = (A_x\\hat{i} + A_y\\hat{j} + A_z\\hat{k}) - (B_x\\hat{i} + B_y\\hat{j} + B_z\\hat{k}) \\\\ = (A_x - B_x)\\hat{i} + (A_y - B_y)\\hat{j} + (A_z - B_z)\\hat{k}[\/latex]<\/p>\n<p style=\"text-align: center\"><span>[latex]\\space=(A_x - B_x)\\underline{\\hat{i}} + (A_y - B_y)\\underline{\\hat{j}} + (A_z - B_z)\\underline{\\hat{k}}[\/latex]\u00a0<\/span><\/p>\n<p><span>This vector equation means we must have simultaneously [latex]A_x-B_x=0[\/latex], [latex]A_y-B_y=0[\/latex], and [latex]A_z-B_z=0[\/latex]<\/span><span>. Hence, we can write [latex]\\vec A=\\vec B[\/latex] <\/span><span>if and only if the corresponding components of vectors\u00a0[latex]\\vec A[\/latex]<\/span><span> and [latex]\\vec B[\/latex]<\/span><span> are equal:<\/span><\/p>\n<p style=\"text-align: center\">[latex]\\vec A =\\vec B[\/latex]\u00a0\u00a0 if \u00a0 [latex]\\begin{bmatrix}A_x=B_x\\\\A_y=B_y\\\\A_z=B_z\\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1167133466190\">Two vectors are equal when their corresponding scalar components are equal.<\/p>\n<p id=\"fs-id1167133321055\">Resolving vectors into their scalar components (i.e., finding their scalar components) and expressing them analytically in vector component form allows us to use vector algebra to find sums or differences of many vectors <em>analytically <\/em>(i.e., using graphical methods). For example, to find the resultant of two vectors [latex]\\overset{\\to }{A}[\/latex]<span>\u00a0<\/span>and<span> [latex]\\vec B[\/latex]<\/span>, we simply add them component by component, as follows:<\/p>\n<p>[latex]\\vec R=\\vec A + \\vec B=(A_x\\underline{\\hat{i}}+A_y\\underline{\\hat{j}}+A_z\\underline{\\hat{k}})+(B_x\\underline{\\hat{i}}+B_y\\underline{\\hat{j}}+B_z\\underline{\\hat{k}})=(A_x+B_x)\\underline{\\hat{i}}+(A_y+B_y)\\underline{\\hat{j}}+(A_z+B_z)\\underline{\\hat{k}}[\/latex]<\/p>\n<p style=\"text-align: left\"><span>In this way,<\/span><span> scalar components of the resultant vector: [latex]\\vec R=(R_x\\underline{\\hat{i}}+R_y\\underline{\\hat{j}}+R_z\\underline{\\hat{k}})[\/latex].<\/span><\/p>\n<p style=\"text-align: center\">[latex]\\begin{matrix}R_x = A_x+B_x\\\\R_y = A_y+B_y\\\\R_z = A_z+B_z\\end{matrix}[\/latex]<\/p>\n<div><\/div>\n<div id=\"fs-id1167132536774\" class=\"unnumbered\">Source: University Physics Volume 1, OpenStax CNX, <a href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-3-algebra-of-vectors\/\">https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-3-algebra-of-vectors\/<\/a><\/div>\n<\/div>\n<div><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-1024x585.png\" alt=\"A campus map with labeled buildings and a yellow path from the FSDE building to the Library, showing vectors R\u2081 to R\u2087, each marked with red points and arrows. X and Y axes are shown near a soccer field.\" class=\"aligncenter wp-image-862 size-large\" width=\"1024\" height=\"585\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-1024x585.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-300x171.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-768x439.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-1536x878.png 1536w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-2048x1170.png 2048w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-65x37.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-225x129.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-27-at-10.52.59-PM-350x200.png 350w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><em>Basically: <\/em>Vectors help describe position, forces, and quantities. Vectors use components, magnitude, and direction (unit vector) to do so.<\/p>\n<p><em>Application: <\/em>A hammock hangs at an angle from the wall. When a person is in the hammock, they are pulling on the wall with a force at an angle. This force vector could be componentized into x and y, using the angle and the weight of the person to calculate it.<\/p>\n<p><em>Looking ahead: <\/em>The next place vectors will appear is in Moments in 1.6.<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":74,"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-156","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/156","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/users\/74"}],"version-history":[{"count":45,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/156\/revisions"}],"predecessor-version":[{"id":2836,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/156\/revisions\/2836"}],"part":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/156\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/media?parent=156"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapter-type?post=156"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/contributor?post=156"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/license?post=156"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}