{"id":206,"date":"2019-08-27T21:54:41","date_gmt":"2019-08-27T21:54:41","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=206"},"modified":"2020-06-18T16:35:51","modified_gmt":"2020-06-18T16:35:51","slug":"elementary-vector-analysis","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/elementary-vector-analysis\/","title":{"rendered":"Elementary Vector Analysis"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Elementary Vector Analysis &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\n$\\newcommand{\\vecb}[1]{{\\bf #1}}\n\\newcommand{\\ihat}{\\hat{\\vecb{i}}}\n\\newcommand{\\jhat}{\\hat{\\vecb{j}}}\n\\newcommand{\\khat}{\\hat{\\vecb{k}}}$\n\n\nIn order to measure many physical quantities, such as force or\nvelocity, we need to determine both a magnitude and a direction.  Such\nquantities are conveniently represented as vectors.\n\n<\/p>\n\n\n\n<p>\n\t\tThe direction of a vector $\\vecb{v}$ in 3-space is specified by its\n\t\tcomponents in the $x$, $y$, and $z$ directions, respectively:\n\t\t$$\n\t\t(x,y,z) \\quad {\\small\\textrm{or}} \\quad x\\ihat + y\\jhat + z\\khat,\n\t\t$$\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"250\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector1.gif?resize=250%2C250&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-430\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\t\t\n\t\t\t\twhere $\\ihat$, $\\jhat$, and $\\khat$ are the <b>coordinate vectors<\/b>\n\t\t\t\talong the $x$, $y$, and $z$-axes.\n\t\t\t\n\t\t\t\n<\/p>\n\n\n\n<p>\n\t\t\t\t\t$\\ihat=(1,0,0)$ <br>\n\t\t\t\t\t$\\jhat=(0,1,0)$ <br>\n\t\t\t\t\t$\\khat=(0,0,1)$\n<\/p>\n\n\n\n<p>\n\t\tThe magnitude of a vector $\\vecb{v}=(x,y,z)$, also called its length or\n\t\t<b>norm<\/b>, is given by\n\t\t$$\n\t\t\\left\\| \\vecb{v} \\right\\| = \\sqrt{x^{2}+y^{2}+z^{2}}.\n\t\t$$\n<\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li>Vectors can be defined in any number of dimensions, though we focus here only on 3-space. <\/li><li>When drawing a vector in 3-space, where you position the vector is unimportant; the vector&#8217;s essential properties are just its magnitude and its direction.  Two vectors are <b>equal<\/b> if and only if corresponding components are equal. <\/li><li>A vector of norm 1 is called a <b>unit vector<\/b>.  The coordinate vectors are examples of unit vectors. <\/li><li>The zero vector, $\\vecb{0} = (0,0,0)$, is the only vector with magnitude 0. <\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Basic Operations on Vectors<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"175\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector2.gif?resize=300%2C175&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-421\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\tTo add or subtract vectors $\\vecb{u} = (u_{1},u_{2},u_{3})$ and\n\t\t$\\vecb{v} = (v_{1},v_{2},v_{3})$, add or subract the corresponding\n\t\tcoordinates:\n\t\t\\begin{eqnarray*}\n\t\t\t\\vecb{u}+\\vecb{v} &amp;= &amp; (u_{1}+v_{1},u_{2}+v_{2},u_{3}+v_{3}) \\\\\n\t\t\t\\vecb{u}-\\vecb{v} &amp;= &amp; (u_{1}-v_{1},u_{2}-v_{2},u_{3}-v_{3}).\n\t\t\\end{eqnarray*}\n<\/p>\n\n\n\n<p> To multiply vector $\\vecb{u}$ by a scalar $k$, multiply each coordinate of $\\vecb{u}$ by $k$: $$ k\\vecb{u}=(ku_{1},ku_{2},ku_{3}). $$ <\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"125\" height=\"150\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector3.gif?resize=125%2C150&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-422\"\/><\/figure><\/div>\n\n\n\n<p>\n\n\t\tThe vector $\\vecb{v}= (2,1,-2) = 2\\ihat + \\jhat -2\\khat$ has\n\t\tmagnitude\n\t\t$$\n\t\t\\left\\| \\vecb{v} \\right\\| = \\sqrt{2^2 +1^2 -(-2)^2} = 3.\n\t\t$$\n<\/p>\n\n\n\n<p>\nThus, the  vector $\\frac{1}{3}\\vecb{v} =\n\\left(\\frac{2}{3},\\frac{1}{3},\\frac{-2}{3}\\right)$ is a unit vector in\nthe same direction as $\\vecb{v}$.\n\n<\/p>\n\n\n\n<p> In general, for $\\vecb{v} \\not= \\vecb{0}$, we can scale (or <b>normalize<\/b>) $\\vecb{v}$ to the unit vector $\\frac{\\vecb{v}}{\\left\\| \\vecb{v} \\right\\|}$ pointing in the same direction as $\\vecb{v}$. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Dot Product<\/h4>\n\n\n\n<p>\n\nLet $\\vecb{u} = (u_{1},u_{2},u_{3})$ and $\\vecb{v} =\n(v_{1},v_{2},v_{3})$.  The <b>dot product<\/b> $\\vecb{u} \\cdot \\vecb{v}$\n(also called the <b>scalar product<\/b> or <b>Euclidean inner\nproduct<\/b>) of $\\vecb{u}$ and $\\vecb{v}$ is defined in two distinct\n(though equivalent) ways:\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"125\" height=\"125\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector4.gif?resize=125%2C125&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-423\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\t\\begin{eqnarray*}\n\t\t\t\\vecb{u} \\cdot \\vecb{v} &amp; = &amp; u_1v_1+u_2v_2+u_3v_3 \\\\\n\t\t\t &amp; = &amp; \\left\\{ \\begin{array}{cl}\n\t\t\t\t\\left\\| \\vecb{u} \\right\\|  \\left\\| \\vecb{v} \\right\\| \\cos \\theta &amp; {\\small\\textrm{if }}\n\t\t\t\t\\vecb{u} \\not= \\vecb{0}, \\vecb{v} \\not= \\vecb{0}\\\\\n\t\t\t\t0 &amp; {\\small\\textrm{if }} \\vecb{u} = \\vecb{0} {\\small\\textrm{ or }} \\vecb{v} = \\vecb{0}\\\\\n\t\t\t\\end{array} \\right.\\\\\n\t\t\t&amp; &amp; \\qquad{\\small\\textrm{where }} 0 \\le \\theta \\le \\pi {\\small\\textrm{ is the angle between }}\n\t\t\t\\vecb{u} {\\small\\textrm{ and }} \\vecb{v} .\n\t\t\\end{eqnarray*}\n<\/p>\n\n\n<style>.kt-accordion-id_6d4a77-62 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_6d4a77-62 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_6d4a77-62 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > 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.kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) 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.kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_6d4a77-62 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_6d4a77-62 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6d4a77-62:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_6d4a77-62 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_6d4a77-62 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_6d4a77-62 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_87c266-2f\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Why are the two definitions equivalent?<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>  $\\newcommand{\\vecb}[1]{{\\bf #1}}$ By the <b><a href=\"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/elementary-vector-analysis\/dot-product\/law-of-cosines\/\">Law of Cosines<\/a><\/b>, for $\\vecb{u} \\not= \\vecb{0}, \\quad \\vecb{v} \\not= \\vecb{0} $, $$ \\left\\| \\vecb{u}-\\vecb{v} \\right\\|^{2} = \\left\\| \\vecb{u} \\right\\|^{2} + \\left\\| \\vecb{v} \\right\\|^{2} &#8211; 2 \\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\|\\cos\\theta . $$ So, for $\\vecb{u} = (u_{1},u_{2},u_{3})$ and $\\vecb{v} = (v_{1},v_{2},v_{3})$, $$ (u_1-v_1)^{2}+(u_2-v_2)^2 + (u_3-v_3)^2 = u_1^2+u_2^2+u_3^2+ v_1^2+v_2^2+v_3^2 -2\\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\|\\cos\\theta . $$ Squaring the expessions on the left and simpliying, \\begin{eqnarray*} -2u_1v_1-2u_2v_2-2u_3v_3 &amp; = &amp; -2\\left\\| \\vecb{u} \\right\\|\\left\\|  \\vecb{v} \\right\\|\\cos\\theta \\\\ u_1v_1 + u_2v_2+u_3v_3 &amp; = &amp; \\left\\| \\vecb{u} \\right\\|\\left\\|  \\vecb{v} \\right\\|\\cos\\theta \\\\ \\vecb{u} \\cdot \\vecb{v} &amp; = &amp; \\left\\| \\vecb{u} \\right\\|\\left\\|  \\vecb{v} \\right\\|\\cos\\theta. \\end{eqnarray*}<\/p>\n\n\n\n<p>\nIf $\\vecb{u} = \\vecb{0}$ or $\\vecb{v} = \\vecb{0}$, both definitions\nimmediately give $\\vecb{u} \\cdot \\vecb{v} = 0$.  Thus, the two\ndefinitions of $\\vecb{u} \\cdot \\vecb{v}$ are equivalent.\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"310\" height=\"260\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector8.gif?resize=310%2C260&#038;ssl=1\" alt=\"Another example of u-v\" class=\"wp-image-427\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Properties of the Dot Product<\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li> $\\vecb{u} \\cdot \\vecb{v} = \\vecb{v} \\cdot \\vecb{u}$ <br> <\/li><li> $\\vecb{u} \\cdot (\\vecb{v} + \\vecb{w}) = (\\vecb{u} \\cdot \\vecb{v}) + (\\vecb{u} \\cdot \\vecb{w})$  <br><\/li><li> $\\vecb{u} \\cdot \\vecb{u} = \\left\\| \\vecb{u} \\right\\|^{2}$ <\/li><\/ul>\n\n\n\n<center>\nSee if you can verify each of these!\n<\/center>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"250\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector5.gif?resize=250%2C250&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-424\"\/><\/figure><\/div>\n\n\n\n<p>\n\n\t\tIf $\\vecb{u}=(1,-2,2)$ and $\\vecb{v}=(-4,0,2)$, then $\\begin{array}{rcl}\n\t\t\t\\vecb{u} \\cdot \\vecb{v} &amp;=&amp; (1)(-4)+(-2)(0)+(2)(2)\\\\\n\t\t\t&amp;=&amp;-1+0+4\\\\\n\t\t\t&amp;=&amp;0.\n\t\t\\end{array}$\n\n\t\t<\/p>\n\n\n\n<p>\n\t\tUsing the second definition of the dot product with $\\left\\| \\vecb{u}\n\t\t\\right\\|=3$ and $\\left\\| \\vecb{v} \\right\\|=2\\sqrt{5}$,\n\t\t$$\n\t\t\\vecb{u} \\cdot \\vecb{v} = 0 = 6\\sqrt{5}\\cos\\theta\n\t\t$$\n\t\tso $\\cos\\theta=0$, yielding $\\theta = \\frac{\\pi}{2}$.\n\n\t\t<\/p>\n\n\n\n<p>\n\t\tThough we might not have guessed it, $\\vecb{u}$ and $\\vecb{v}$ are\n\t\tperpendicular to each other!\n\n\t\t<\/p>\n\n\n\n<p>\n\t\tIn general, \n<\/p>\n\n\n\n<p>\n$\\qquad$ Two non-zero vectors $\\vecb{u}$ and $\\vecb{v}$ are perpendicular (or \n<b>orthogonal<\/b>) if and only if $\\vecb{u} \\cdot \\vecb{v} = 0$.\n\n<\/p>\n\n\n<style>.kt-accordion-id_b75d81-1c .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_b75d81-1c .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_b75d81-1c > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_b75d81-1c > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_b75d81-1c .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_b75d81-1c .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_b75d81-1c > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b75d81-1c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_b75d81-1c .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_b75d81-1c .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_b75d81-1c kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_cc2191-0a\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Dot Zero &#8211; Proof<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-media-text alignwide has-media-on-the-right\"><figure class=\"wp-block-media-text__media\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"200\" height=\"125\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector9.gif?resize=200%2C125&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-428\"\/><\/figure><div class=\"wp-block-media-text__content\">\n<p>\n\nLet $\\vecb{u}$ and $\\vecb{v}$ be non-zero vectors.  Suppose $\\vecb{u}$ is orthogonal to $\\vecb{v}$.  Then the angle between $\\vecb{u}$ and $\\vecb{v}$ is $\\frac{\\pi}{2}$.  Thus,  \\begin{eqnarray*} \\vecb{u} \\cdot \\vecb{v}  &amp; = &amp; \\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\| \\cos \\frac{\\pi}{2} \\\\  &amp; = &amp; \\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\| (0)\\\\  &amp; = &amp; 0. \\end{eqnarray*} \n\n<\/p>\n<\/div><\/div>\n\n\n\n<p>\nSuppose now that $\\vecb{u} \\cdot \\vecb{v} = 0$.  Then \n\\begin{eqnarray*}\n\t\\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\| \\cos\\theta &amp; = &amp; 0\\\\\n\t\\cos\\theta &amp; = &amp; \\frac{0}{\\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v}\n\t\\right\\|} \\\\\n\t\\cos\\theta &amp; = &amp; 0.\n\\end{eqnarray*}\n\n<\/p>\n\n\n\n<p>\nSo $\\theta = \\frac{\\pi}{2}$.  Thus $\\vecb{u}$ is orthogonal to $\\vecb{v}$.\n\n<!------------------------>\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Projection of a Vector<\/h4>\n\n\n\n<p>\n\t\tIt is often useful to resolve a vector $\\vecb{v}$ into the sum of\n\t\tvector components parallel and perpendicular to a vector $\\vecb{u}$.\n\n\t\t<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"200\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector6.gif?resize=250%2C200&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-425\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\tConsider first the parallel component, which is called the \n\t\t<b>projection of $\\vecb{v}$ onto $\\vecb{u}$<\/b>.  This projection should be in\n\t\tthe direction of $\\vecb{u}$ and should have magnitude $\\left\\| \\vecb{v}\n\t\t\\right\\|\\cos\\theta$, where $0 \\le \\theta \\le \\pi$ is the angle between\n\t\t$\\vecb{u}$ and $\\vecb{v}$.  Let&#8217;s normalize  $\\vecb{u}$ to\n\t\t$\\frac{\\vecb{u}}{\\left\\| \\vecb{u} \\right\\|}$ and then scale this by\n\t\tthe magnitude $\\left\\| \\vecb{v} \\right\\|\\cos\\theta$:\n\n\t\t<\/p>\n\n\n\n<p>\n\t\t\t\t$\\begin{array}{rl}\n\t\t\t\t\t = &amp; \\left(\\left\\| \\vecb{v} \\right\\|\\cos\\theta\\right)\\frac{\\vecb{u}}{\\left\\|\n\t\t\t\t\t\\vecb{u} \\right\\|} \\\\\n\t\t\t\t\t = &amp; \\frac{\\left\\| \\vecb{v} \\right\\|\\left\\|\n\t\t\t\t\t\\vecb{u}\\right\\|\\cos\\theta}{\\left\\| \\vecb{u} \\right\\|^{2}}\\vecb{u}\\\\\n\t\t\t\t\t = &amp;  \\frac{\\vecb{v} \\cdot \\vecb{u}}{\\left\\| \\vecb{u} \\right\\|^{2}}\\vecb{u}.\n\t\t\t\t\\end{array}$\n<\/p>\n\n\n\n<p>\nThe perpendicular vector component of $\\vecb{v}$ is then just the\ndifference between $\\vecb{v}$ and the projection of $\\vecb{v}$ onto\n$\\vecb{u}$.\n\n<\/p>\n\n\n\n<p>\nIn summary,\n\n<\/p>\n\n\n\n<p> projection of $\\vecb{v}$ onto $\\vecb{u}$ <\/p>\n\n\n\n<p> vector component of $\\vecb{v}$ perpendicular to $\\vecb{u}$ <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Cross Product<\/h4>\n\n\n\n<p>\n\nLet $\\vecb{u} = (u_{1},u_{2},u_{3})$ and $\\vecb{v} =\n(v_{1},v_{2},v_{3})$.  The <b>cross product<\/b> $\\vecb{u} \\times\n\\vecb{v}$ yields a vector perpendicular to both $\\vecb{u}$ and $\\vecb{v}$\nwith direction determined by the right-hand rule.  Specifically, \n$$\n\\vecb{u} \\times \\vecb{v} = (u_{2}v_3-u_3v_2)\\ihat &#8211;\n(u_1v_3-u_3v_1)\\jhat + (u_1v_2-u_2v_1)\\khat.\n$$\nIt can also be shown that\n$$\n\\left\\| \\vecb{u} \\times \\vecb{v} \\right\\| = \\left\\| \\vecb{u}\n\\right\\|\\left\\| \\vecb{v} \\right\\|\\sin\\theta \\quad {\\small\\textrm{for }} \\vecb{u}\n\\not= \\vecb{0}, \\quad \\vecb{v} \\not= \\vecb{0}\n$$\n<\/p>\n\n\n<style>.kt-accordion-id_f84ee6-a0 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_f84ee6-a0 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_f84ee6-a0 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_f84ee6-a0 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_f84ee6-a0 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_f84ee6-a0 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_f84ee6-a0 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_f84ee6-a0:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_f84ee6-a0 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_f84ee6-a0 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_f84ee6-a0 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_263610-43\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Cross Product &#8211; Proof<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n\nLet $\\vecb{u}$ and $\\vecb{v}$ be non-zero vectors. We will use Lagrange&#8217;s Identity, which says that $$ \\left\\| \\vecb{u} \\times \\vecb{v} \\right\\|^{2}  = \\left\\| \\vecb{u} \\right\\|^{2}\\left\\| \\vecb{v} \\right\\|^{2} &#8211; \\left( \\vecb{u} \\cdot \\vecb{v} \\right)^{2}. $$ Since $\\vecb{u} \\cdot \\vecb{v} =\\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\| \\cos\\theta$, \\begin{eqnarray*} \\left\\| \\vecb{u} \\times \\vecb{v} \\right\\|^{2}  &amp;= &amp;\\left\\| \\vecb{u} \\right\\|^{2}\\left\\| \\vecb{v} \\right\\|^{2}-\\left( \\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\| \\cos\\theta \\right)^{2} \\\\  &amp; = &amp;  \\left\\| \\vecb{u} \\right\\|^{2}\\left\\| \\vecb{v} \\right\\|^{2} &#8211;  \\left\\| \\vecb{u} \\right\\|^{2}\\left\\| \\vecb{v} \\right\\|^{2}  \\cos^{2}\\theta\\\\  &amp; = &amp; \\left\\| \\vecb{u} \\right\\|^{2}\\left\\| \\vecb{v} \\right\\|^{2} \\left  ( 1 &#8211; \\cos^{2}\\theta \\right) \\\\  &amp; = &amp; \\left\\| \\vecb{u} \\right\\|^{2}\\left\\| \\vecb{v} \\right\\|^{2} \\sin^{2}\\theta. \\end{eqnarray*} So $$ \\left\\| \\vecb{u} \\times \\vecb{v} \\right\\| = \\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\|\\sin\\theta.  $$  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p> <br>Thus, the magnitude $\\left\\| \\vecb{u} \\times \\vecb{v} \\right\\|$ gives the area of the parallelogram formed by $\\vecb{u}$ and $\\vecb{v}$. <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"310\" height=\"260\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/vector7.gif?resize=310%2C260&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-426\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\tAs implied by the geometric interpretation, \n\t\t<\/p>\n\n\n\n<center>\n\t\t\t<p>\n<\/p><p align=\"center\">\n\t\t\t\tNon zero vectors $\\vecb{u}$ and $\\vecb{v}$ are\n\t\t\t\tparallel if and only if $\\vecb{u} \\times \\vecb{v}=\\vecb{0}$. \n<\/p>\n\n\n<style>.kt-accordion-id_9c481e-27 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_9c481e-27 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_9c481e-27 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_9c481e-27 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_9c481e-27 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_9c481e-27 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_9c481e-27 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9c481e-27:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_9c481e-27 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_9c481e-27 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_9c481e-27 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_428f3a-de\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Cross Zero &#8211; Proof<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n<!------------------------>\n\nLet $\\vecb{u}$ and $\\vecb{v}$ be non-zero vectors.\n\n<\/p>\n\n\n\n<p>\nSuppose $\\vecb{u}$ is parallel to $\\vecb{v}$.  Then the angle $\\theta$\nis 0 of $\\pi$.  So\n\\begin{eqnarray*}\n\t\\left\\| \\vecb{u} \\times \\vecb{v} \\right\\| &amp; = &amp;  \\left\\| \\vecb{u}\n\t\\right\\|\\left\\| \\vecb{v} \\right\\| \\sin\\theta \\\\\n\t &amp; = &amp;  \\left\\| \\vecb{u}\n\t\\right\\|\\left\\| \\vecb{v} \\right\\| (0)\\\\\n\t &amp; = &amp; 0.\n\\end{eqnarray*}\nThus $\\vecb{u} \\times \\vecb{v} = \\vecb{0}$.\n\n<\/p>\n\n\n\n<p> Suppose now that $\\vecb{u} \\times \\vecb{v} = \\vecb{0}$.  Then  \\begin{eqnarray*} \\left\\| \\vecb{u} \\times \\vecb{v} \\right\\| &amp; = &amp; 0 \\\\ \\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\| \\sin\\theta &amp; = &amp; 0 \\\\ \\sin\\theta &amp; = &amp; \\frac{0}{\\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\|}  \\\\ \\sin\\theta &amp; = &amp; 0. \\end{eqnarray*} So $\\theta = 0$ or $\\pi$.  In either case, $\\vecb{u}$ is parallel to $\\vecb{v}$.  <br><\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Properties of the Cross Product<\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li> $\\vecb{u} \\times \\vecb{v} = &#8211; \\left( \\vecb{v} \\times \\vecb{u}\n\t\t\t\t\\right)$\n\n\t\t\t<br><br>\n\t\t\t<\/li><li> $\\vecb{u} \\times \\left( \\vecb{v} + \\vecb{w} \\right) = \\left(\\vecb{u}\n\t\t\t\t\\times \\vecb{v} \\right) + \\left( \\vecb{u} \\times \\vecb{w} \\right) $\n\n\t\t\t<br><br>\n\t\t\t<\/li><li> $\\vecb{u} \\times \\vecb{u} = \\vecb{0}$\n\t\t<\/li><\/ul>\n\n\n\n<p>\n\t\tAgain, see if you can verify each of these.\n<\/p>\n\n\n<style>.kt-accordion-id_9de9f6-b7 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_9de9f6-b7 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_9de9f6-b7 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_9de9f6-b7 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_9de9f6-b7 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( 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.wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_9de9f6-b7:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_9de9f6-b7 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_9de9f6-b7 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_9de9f6-b7 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_bd0a5c-cb\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Connections Between the Dot and Cross Product<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<ul class=\"wp-block-list\"><li> <strong>Lagrange&#8217;s Identity<\/strong> $$ \\left\\| \\vecb{u} \\times \\vecb{v} \\right\\|^{2}  = \\left\\| \\vecb{u} \\right\\|^{2}\\left\\| \\vecb{v} \\right\\|^{2} &#8211; \\left( \\vecb{u} \\cdot \\vecb{v} \\right)^{2}. $$ <br><br> <\/li><li> <strong>Volume of a Parallelepiped<\/strong>  Consider the parallelepiped with adjacent sides $\\vecb{a}$, $\\vecb{b}$, and $\\vecb{c}$.  The area of the base is $\\left\\|\\vecb{b}\\times\\vecb{c}\\right\\|$.    The height of the parallelepiped is $\\left\\|\\vecb{a}\\right\\|\\cos\\theta$.  Thus, the parallelepiped has volume $$ V = \\left\\|\\vecb{b}\\times\\vecb{c}\\right\\|\\left\\|\\vecb{a}\\right\\|\\cos\\theta. $$  But recall that $\\vecb{u} \\cdot \\vecb{v} =\\left\\|\\vecb{u}\\right\\|\\left\\|\\vecb{v}\\right\\|\\cos\\theta$ for $ \\vecb{u} \\not= \\vecb{0}, \\quad \\vecb{v} \\not= \\vecb{0}$.  <br> Thus,  $$ V = \\vecb{a} \\cdot \\left( \\vecb{b} \\times \\vecb{c} \\right). $$   It doesn&#8217;t matter which two vectors define the &#8220;base&#8221; of the parallelepiped: <br>  $\\vecb{a} \\cdot \\left( \\vecb{b} \\times \\vecb{c} \\right) =$ <br>  $\\vecb{c} \\cdot \\left( \\vecb{a} \\times \\vecb{b} \\right) =$ <br> $\\vecb{b} \\cdot \\left( \\vecb{c} \\times \\vecb{a} \\right)$. <\/li><\/ul>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<p>\n\nLet $\\vecb{u} = (u_{1},u_{2},u_{3})$ and $\\vecb{v} =\n(v_{1},v_{2},v_{3})$.\n\n\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> <b>Basic Operations, Norm of a vector<\/b> <p> \\begin{eqnarray*} \\vecb{u}+\\vecb{v} &amp;= &amp; (u_{1}+v_{1},u_{2}+v_{2},u_{3}+v_{3}) \\\\ \\vecb{u}-\\vecb{v} &amp;= &amp; (u_{1}-v_{1},u_{2}-v_{2},u_{3}-v_{3}) \\\\ k\\vecb{u} &amp; = &amp; (ku_{1},ku_{2},ku_{3})  \\\\ \\left\\| \\vecb{v} \\right\\| &amp; = &amp; \\sqrt{x^{2}+y^{2}+z^{2}}   \\end{eqnarray*} <br> <\/p><\/li><li> <b>Dot Product<\/b> <p> \\begin{eqnarray*} \\vecb{u} \\cdot \\vecb{v} &amp; = &amp; u_1v_1+u_2v_2+u_3v_3 \\\\  &amp; = &amp; \\left\\{ \\begin{array}{cl} \\left\\| \\vecb{u} \\right\\|  \\left\\| \\vecb{v} \\right\\| \\cos \\theta &amp; {\\small\\textrm{if }} \\vecb{u} \\not= \\vecb{0}, \\vecb{v} \\not= \\vecb{0}\\\\ 0 &amp; {\\small\\textrm{if }} \\vecb{u} = \\vecb{0} {\\small\\textrm{ or }} \\vecb{v} = \\vecb{0}\\\\ \\end{array} \\right.\\\\ &amp; &amp; \\qquad{\\small\\textrm{where }} 0 \\le \\theta \\le \\pi {\\small\\textrm{ is the angle between }} \\vecb{u} {\\small\\textrm{ and }} \\vecb{v}  \\end{eqnarray*}  $\\qquad$ For $\\vecb{u} \\not= \\vecb{0}, \\quad \\vecb{v} \\not= \\vecb{0}$,  $\\qquad\\qquad \\vecb{u} \\cdot \\vecb{v} = 0$ if and only if $\\vecb{u}$ is orthogonal to $\\vecb{v}$. <br><\/p><\/li><li> <b>Projection of a Vector<\/b> <p> <\/p><p align=\"center\">    projection of $\\vecb{v}$ onto $\\vecb{u}$      vector component of $\\vecb{v}$ perpendicular to $\\vecb{u}$   <\/p> <\/li><li> <b>Cross Product<\/b> <p> \\begin{eqnarray*} \\vecb{u} \\times \\vecb{v} &amp; = &amp; (u_{2}v_3-u_3v_2)\\ihat &#8211; (u_1v_3-u_3v_1)\\jhat + (u_1v_2-u_2v_1)\\khat\\\\ \\left\\| \\vecb{u} \\times \\vecb{v} \\right\\| &amp; = &amp; \\left\\| \\vecb{u} \\right\\|\\left\\| \\vecb{v} \\right\\|\\sin\\theta \\quad {\\small\\textrm{for }} \\vecb{u} \\not= \\vecb{0}, \\quad \\vecb{v} \\not= \\vecb{0} \\end{eqnarray*} <\/p><p align=\"center\">  <\/p> <\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ2310\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Elementary Vector Analysis &#8211; HMC Calculus Tutorial $\\newcommand{\\vecb}[1]{{\\bf #1}} \\newcommand{\\ihat}{\\hat{\\vecb{i}}} \\newcommand{\\jhat}{\\hat{\\vecb{j}}} \\newcommand{\\khat}{\\hat{\\vecb{k}}}$ In order to measure many physical quantities, such as force or velocity, we need to determine both a magnitude and a direction. Such quantities are conveniently represented as vectors. The direction of a vector $\\vecb{v}$ in 3-space is specified by its components in&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":59,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-206","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/206","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=206"}],"version-history":[{"count":13,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/206\/revisions"}],"predecessor-version":[{"id":1227,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/206\/revisions\/1227"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/59"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=206"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}