{"id":208,"date":"2019-08-27T23:16:16","date_gmt":"2019-08-27T23:16:16","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=208"},"modified":"2019-12-05T00:37:07","modified_gmt":"2019-12-05T00:37:07","slug":"lines-planes-and-vectors","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/lines-planes-and-vectors\/","title":{"rendered":"Lines, Planes, and Vectors"},"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>Lines, Planes, and Vectors &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>  $ \\newcommand{\\vecb}[1]{{\\bf #1}} $In this tutorial, we will use vector methods to represent lines and planes in 3-space. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Displacement Vector<\/h4>\n\n\n\n<p>\n\nThe displacement vector $\\vecb{v}$ with initial point $(x_{1},y_{1},z_{1})$ and\nterminal point $(x_{2},y_{2},z_{2})$ is\n$$\n\\vecb{v}=(x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1}).\n$$\n\n<\/p>\n\n\n<style>.kt-accordion-id_390d99-7f .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_390d99-7f .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_390d99-7f > .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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f > .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_390d99-7f .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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_390d99-7f > .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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f: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_390d99-7f .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_390d99-7f .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_390d99-7f 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_651e4a-80\"><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?<\/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\n$ \\newcommand{\\vecb}[1]{{\\bf #1}} $\n\nConsider the displacement vector $\\vecb{v}$ with initial point\n$(x_{1},y_{1},z_{1})$ and \nterminal point $(x_{2},y_{2},z_{2})$.\n\n<\/p>\n\n\n\n<p>\nSince $\\vecb{v}=\\vecb{u}-\\vecb{w}$,\n\n\\begin{eqnarray*}\n\t\\vecb{v} &amp;= &amp; (x_{2},y_{2},z_{2})- (x_{1},y_{1},z_{1}) \\\\\n\t &amp; = &amp;  (x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1}).\n\\end{eqnarray*}\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p> <br>That is, if vector $\\vecb{v}$ were positioned with its initial point at the origin, then its terminal point would be at $(x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1})$. <\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nThe vector $\\vecb{v}$ with initial point $(-1,4,5)$ and final point\n$(4,-3,2)$ is\n$$\n\\vecb{v} = \\left( 4-(-1),-3-4,2-5 \\right) = (5,-7,-3).\n$$\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Parametric Equations for a Line in 3-space<\/h4>\n\n\n\n<p>\n\nThe line throught the point $(x_{0},y_{0},z_{0})$ and parallel to the\nnon-zero vector $\\vecb{v} = (a,b,c)$ has parametric equations\n\n\\begin{eqnarray*}\n\tx &amp; = &amp; x_{0} + at  \\\\\n\ty &amp; = &amp; y_{0} + bt  \\\\\n\tz &amp; = &amp; z_{0} + ct.\n\\end{eqnarray*}\n\n<\/p>\n\n\n<style>.kt-accordion-id_267620-a2 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_267620-a2 .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_267620-a2 > .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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2 > .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_267620-a2 .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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_267620-a2 > .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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2: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_267620-a2 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_267620-a2 .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_267620-a2 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_ccf187-89\"><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?<\/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}} $ Consider the line through the point $(x_{0},y_{0},z_{0})$ and parallel to the non-zero vector $\\vecb{v}~=~(a,b,c)$.  A point $(x,y,z)$ is on the line if and only if the displacement vector with initial point $(x_{0},y_{0},z_{0})$ and final point $(x,y,z)$ is parallel to $\\vecb{v}$.  That is, $(x-x_{0},y-y_{0},z-z_{0})$ must be a scalar multiple of $\\vecb{v}$: $$ (x-x_{0},y-y_{0},z-z_{0})=t(a,b,c). $$ Componentwise, \\begin{eqnarray*} x &#8211; x_{0} &amp; = &amp; at  \\\\ y &#8211; y_{0} &amp; = &amp; bt  \\\\ z &#8211; z_{0} &amp; = &amp; ct. \\end{eqnarray*} Thus, we obtain the parametric equations \\begin{eqnarray*} x &amp; = &amp; x_{0} + at  \\\\ y &amp; = &amp; y_{0} + bt  \\\\ z &amp; = &amp; z_{0} + ct. \\end{eqnarray*}   <br><\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>The line through $(2,-1,3)$ and parallel to the vector $\\vecb{v}=(3,-7,4)$ has parametric equations \\begin{eqnarray*} x &amp;= &amp; 2+3t  \\\\ y &amp; =&amp; -1-7t \\\\ z &amp; =&amp; 3+4t. \\end{eqnarray*} Notice that when $t=0$, we are at the point $(2,-1,3)$.  As $t$ increases or decreases from 0, we move away from this point parallel to the direction indicated by $(3,-7,4)$. <\/p>\n\n\n\n<p>\nIf you know two points $p_{1} = (x_{1},y_{1},z_{1})$ and\n$p_{2}=(x_{2},y_{2},z_{2})$ that a line passes through, you can find a\nparametrization for the lilne.  First, find the displacement vector\n$\\vecb{v}=(x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1})$.  then write down\nparametric equations for the line through either $p_{1}$ or $p_{2}$\nand parallel to $\\vecb{v}$.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Equation of a Plane in 3-space<\/h4>\n\n\n\n<p>\n\nThe equation of the plane containing the point $(x_{0},y_{0},z_{0})$\nwith normal vector $\\vecb{n} = (a,b,c)$ is \n$$\na(x-x_{0})+ b(y-y_{0})+c(z-z_{0})=0.\n$$\n\n<\/p>\n\n\n<style>.kt-accordion-id_5dbd1b-54 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_5dbd1b-54 .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_5dbd1b-54 > .kt-accordion-inner-wrap > 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.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_5dbd1b-54: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_5dbd1b-54: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_5dbd1b-54 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_5dbd1b-54 .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_5dbd1b-54 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_617f7e-96\"><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?<\/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\n$ \\newcommand{\\vecb}[1]{{\\bf #1}} $\n\nConsider the plane containing the point $(x_{0},y_{0},z_{0})$ with the\nnormal vector $\\vecb{n} = (a,b,c)$.  A point $(x,y,z)$ is in the plane\nif and only if the displacement vector with initial point\n$(x_{0},y_{0},z_{0})$ and final point $(x,y,z)$ is perpendicular to\n$\\vecb{n}$.  That is,\n\n<\/p>\n\n\n\n<p>\n\t\t\\begin{eqnarray*}\n\t\t\t(a,b,c) \\cdot (x-x_{0},y-y_{0},z-z_{0}) &amp; = &amp; 0 \\\\\n\t\t\ta(x-x_{0})+ b(y-y_{0})+c(z-z_{0}) &amp; = &amp; 0. \n\t\t\\end{eqnarray*}\n<\/p>\n\n\n\n<p>\n\t\t\tRecall that vectors $\\vecb{u}$ and $\\vecb{v}$ are perpendicular if and only if $$\\vecb{u} \\cdot \\vecb{v} = 0.$$\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p> <br>Thus, the graph of the equation $$ ax+by+cz=d $$ is a plane with normal vector $ (a,b,c)$. <\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nThe equation of the plane containing $(2,4,-1)$ and normal to the\nvector $\\vecb{n} = (3,5,-2)$ is \n$$\n3(x-2)+5(y-4)-2(z-(-1))=0.\n$$\nSimplifying,\n$$\n3x+5y-2z=28.\n$$\n\nWith a little extra work, we can use this procedure to find the\nequation of the plane defined by any thee points.  First, compute\ndisplacement vectors $\\vecb{u}$ and $\\vecb{v}$ between two pairs of\nthese points.  Then $\\vecb{n} = \\vecb{u} \\times \\vecb{v}$ in normal to\nthe plane.  Now, use one of the points and the vector $\\vecb{n} =\n\\vecb{u} \\times \\vecb{v}$ to obtain the equation of the plane.\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<h4>Key Concepts<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> <b>Displacement Vector<\/b>\n\t\t<p>\n\t\tThe displacement vector $\\vecb{v}$ with initial point\n\t\t$(x_{1},y_{1},z_{1})$ and terminal point $(x_{2},y_{2},z_{2})$ is $\\vecb{v}=(x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1})$.\n\t<br><br>\n\t<\/p><\/li><li> <b>Parametric Equations for a line in 3-space<\/b>\n\t\t<p>\n\t\tThe line throught he point $(x_{0},y_{0},z_{0})$ and parallel to the\n\t\tnon-zero vector $\\vecb{v} = (a,b,c)$ has parametric equations\n\n\t\t\\begin{eqnarray*}\n\t\t\tx &amp; = &amp; x_{0} + at  \\\\\n\t\t\ty &amp; = &amp; y_{0} + bt  \\\\\n\t\t\tz &amp; = &amp; z_{0} + ct.\n\t\t\\end{eqnarray*}\n\t<br><br>\n\t<\/p><\/li><li> <b>Equation of a plane in 3-space<\/b>\n\t\t<p>\n\t\tThe equation of the plane containing the point $(x_{0},y_{0},z_{0})$\n\t\twith normal vector $\\vecb{n} = (a,b,c)$ is \n\t\t$$\n\t\ta(x-x_{0})+ b(y-y_{0})+c(z-z_{0})=0.\n\t\t$$\n\n<\/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\/QZ2110\/\">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>Lines, Planes, and Vectors &#8211; HMC Calculus Tutorial $ \\newcommand{\\vecb}[1]{{\\bf #1}} $In this tutorial, we will use vector methods to represent lines and planes in 3-space. Displacement Vector The displacement vector $\\vecb{v}$ with initial point $(x_{1},y_{1},z_{1})$ and terminal point $(x_{2},y_{2},z_{2})$ is $$ \\vecb{v}=(x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1}). $$ That is, if vector $\\vecb{v}$ were positioned with its initial point&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-208","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/208","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=208"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/208\/revisions"}],"predecessor-version":[{"id":1135,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/208\/revisions\/1135"}],"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=208"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}