{"id":530,"date":"2025-10-09T15:18:48","date_gmt":"2025-10-09T15:18:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=530"},"modified":"2025-10-22T14:23:23","modified_gmt":"2025-10-22T14:23:23","slug":"lines-and-planes-in-space-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/lines-and-planes-in-space-learn-it-1\/","title":{"raw":"Lines and Planes in Space: Learn It 1","rendered":"Lines and Planes in Space: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write different types of equations (vector, parametric, symmetric) to describe lines in space<\/li>\r\n \t<li>Find the distance from a point to a given line<\/li>\r\n \t<li>Write equations for planes using vectors and scalars<\/li>\r\n \t<li>Find the angle between two planes<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Equations for a Line in Space<\/h2>\r\n<p class=\"whitespace-normal break-words\">You're already comfortable writing equations for lines in two dimensions. In three dimensions, the concept is similar, but instead of using slope to describe direction, we use a <strong>direction vector<\/strong> parallel to the line.<\/p>\r\n<p class=\"whitespace-normal break-words\">Two vectors are parallel when they point in the same or opposite directions. For nonzero vectors [latex]\\textbf{u}[\/latex] and [latex]\\textbf{v}[\/latex], this means there exists a scalar [latex]k[\/latex] such that [latex]\\textbf{u} = k\\textbf{v}[\/latex].<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>parallel vectors<\/h3>\r\nTwo nonzero vectors [latex]\\textbf{u}[\/latex] and [latex]\\textbf{v}[\/latex] are parallel if and only if [latex]\\textbf{u} = k\\textbf{v}[\/latex] for some scalar [latex]k[\/latex]. When [latex]k &gt; 0[\/latex], the vectors have the same direction. When [latex]k &lt; 0[\/latex], they have opposite directions.\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The zero vector [latex]\\textbf 0[\/latex] is considered to be parallel to all vectors.<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Vector Equation of a Line<\/h3>\r\n<p class=\"whitespace-normal break-words\">Consider a line [latex]L[\/latex] passing through point [latex]P(x_0, y_0, z_0)[\/latex] with direction vector [latex]\\textbf{v} = \\langle a, b, c \\rangle[\/latex]. For any point [latex]Q(x, y, z)[\/latex] on this line, the vector [latex]\\overrightarrow{PQ}[\/latex] must be parallel to [latex]\\textbf{v}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">This means [latex]\\overrightarrow{PQ} = t\\textbf{v}[\/latex] for some scalar [latex]t[\/latex]. Since [latex]\\overrightarrow{PQ} = \\langle x - x_0, y - y_0, z - z_0 \\rangle[\/latex], we have:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\langle x - x_0, y - y_0, z - z_0 \\rangle = t\\langle a, b, c \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Using vector addition, we can rewrite this as:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\langle x, y, z \\rangle = \\langle x_0, y_0, z_0 \\rangle + t\\langle a, b, c \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Setting [latex]\\textbf{r} = \\langle x, y, z \\rangle[\/latex] and [latex]\\textbf{r}_0 = \\langle x_0, y_0, z_0 \\rangle[\/latex], we get the <strong>vector equation of a line<\/strong>:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\textbf{r} = \\textbf{r}_0 + t\\textbf{v}[\/latex]<\/p>\r\n\r\n[caption id=\"attachment_5157\" align=\"aligncenter\" width=\"433\"]<img class=\"wp-image-5157 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/27211934\/2.63.jpg\" alt=\"\" width=\"433\" height=\"390\" \/> Figure 1. Vector [latex]\\textbf v[\/latex] is the direction vector for [latex]\\overrightarrow{PQ}[\/latex].[\/caption]\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Parametric and Symmetric Equations<\/h3>\r\n<p class=\"whitespace-normal break-words\">From the vector equation, we can extract three component equations:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x - x_0 = ta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y - y_0 = tb[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z - z_0 = tc[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Solving for [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex] gives us the <strong>parametric equations<\/strong>:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]x = x_0 + ta, \\quad y = y_0 + tb, \\quad z = z_0 + tc[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">If [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are all nonzero, we can solve each equation for [latex]t[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\frac{x - x_0}{a} = t, \\quad \\frac{y - y_0}{b} = t, \\quad \\frac{z - z_0}{c} = t[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Since all expressions equal [latex]t[\/latex], we can set them equal to create <strong>symmetric equations<\/strong>:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\frac{x - x_0}{a} = \\frac{y - y_0}{b} = \\frac{z - z_0}{c}[\/latex]<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>line equations in three dimensions<\/h3>\r\n<p class=\"whitespace-normal break-words\">A line [latex]L[\/latex] through point [latex]P(x_0, y_0, z_0)[\/latex] parallel to vector [latex]\\textbf{v} = \\langle a, b, c \\rangle[\/latex] can be described by:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vector form<\/strong>: [latex]\\textbf{r} = \\textbf{r}_0 + t\\textbf{v}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parametric form<\/strong>: [latex]x = x_0 + ta, \\quad y = y_0 + tb, \\quad z = z_0 + tc[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Symmetric form<\/strong>: [latex]\\frac{x - x_0}{a} = \\frac{y - y_0}{b} = \\frac{z - z_0}{c}[\/latex] (when [latex]a, b, c \\neq 0[\/latex])<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0The parametric and symmetric equations of a line are not unique. Using a different direction vector or different point on the line creates equivalent but different-looking equations that describe the same line.<\/section><section class=\"textbox example\" aria-label=\"Example\">Find parametric and symmetric equations of the line passing through points [latex](1, 4, -2)[\/latex] and\u00a0[latex](-3, 5, 0)[\/latex].[reveal-answer q=\"389475782\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"389475782\"]\r\n<p id=\"fs-id1163723796963\">First, identify a vector parallel to the line:<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\bf{v}}=\\langle-3-1,5-4,0-(-2)\\rangle=\\langle-4,1,2\\rangle[\/latex].<\/p>\r\n<p id=\"fs-id1163724059770\">Use either of the given points on the line to complete the parametric equations:<\/p>\r\n<p style=\"text-align: center;\">[latex]x=1-4t[\/latex],\u00a0[latex]y=4+t[\/latex], and\u00a0[latex]z=-2+2t[\/latex].<\/p>\r\n<p id=\"fs-id1163723281292\">Solve each equation for [latex]t[\/latex] to create the symmetric equation of the line:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{x-1}{-4}=y-4=\\frac{z+2}2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3>Line Segments<\/h3>\r\n<p class=\"whitespace-normal break-words\">Sometimes you need equations for just a portion of a line, not the entire line. This requires restricting the parameter [latex]t[\/latex] to a specific interval.<\/p>\r\n<p class=\"whitespace-normal break-words\">Consider two points [latex]P(x_0, y_0, z_0)[\/latex] and [latex]Q(x_1, y_1, z_1)[\/latex] with position vectors [latex]\\textbf{p} = \\langle x_0, y_0, z_0 \\rangle[\/latex] and [latex]\\textbf{q} = \\langle x_1, y_1, z_1 \\rangle[\/latex]. To find the equation of the line segment between these points, we use [latex]P[\/latex] as our reference point and [latex]\\overrightarrow{PQ} = \\langle x_1 - x_0, y_1 - y_0, z_1 - z_0 \\rangle[\/latex] as our direction vector.<\/p>\r\n<p class=\"whitespace-normal break-words\">Starting with the vector equation [latex]\\textbf{r} = \\textbf{p} + t(\\overrightarrow{PQ})[\/latex], we can manipulate this algebraically:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} \\textbf{r} &amp;= \\textbf{p} + t(\\overrightarrow{PQ}) \\\\ &amp;= \\langle x_0, y_0, z_0 \\rangle + t\\langle x_1 - x_0, y_1 - y_0, z_1 - z_0 \\rangle \\\\ &amp;= \\langle x_0, y_0, z_0 \\rangle + t(\\langle x_1, y_1, z_1 \\rangle - \\langle x_0, y_0, z_0 \\rangle) \\\\ &amp;= \\langle x_0, y_0, z_0 \\rangle + t\\langle x_1, y_1, z_1 \\rangle - t\\langle x_0, y_0, z_0 \\rangle \\\\ &amp;= (1-t)\\langle x_0, y_0, z_0 \\rangle + t\\langle x_1, y_1, z_1 \\rangle \\\\ &amp;= (1-t)\\textbf{p} + t\\textbf{q} \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This gives us the <strong>vector equation for a line segment<\/strong>: [latex]\\textbf{r} = (1-t)\\textbf{p} + t\\textbf{q}[\/latex]<\/p>\r\nWe can also derive the parametric equations directly:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n\\textbf{r} &amp;= \\textbf{p} + t(\\overrightarrow{PQ}) \\\\\r\n\\langle x, y, z \\rangle &amp;= \\langle x_0, y_0, z_0 \\rangle + t\\langle x_1 - x_0, y_1 - y_0, z_1 - z_0 \\rangle \\\\\r\n&amp;= \\langle x_0 + t(x_1 - x_0), y_0 + t(y_1 - y_0), z_0 + t(z_1 - z_0) \\rangle\r\n\\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Notice what happens at the endpoints:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">When [latex]t = 0[\/latex]: [latex]\\textbf{r} = \\textbf{p}[\/latex] (we're at point [latex]P[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When [latex]t = 1[\/latex]: [latex]\\textbf{r} = \\textbf{q}[\/latex] (we're at point [latex]Q[\/latex])<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>line segment equation<\/h3>\r\n<p class=\"whitespace-normal break-words\">The line segment between points [latex]P[\/latex] and [latex]Q[\/latex] is described by:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vector form<\/strong>: [latex]\\textbf{r} = (1-t)\\textbf{p} + t\\textbf{q}, \\quad 0 \\leq t \\leq 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parametric form<\/strong>: [latex]x = x_0 + t(x_1 - x_0)[\/latex], [latex]y = y_0 + t(y_1 - y_0)[\/latex], [latex]z = z_0 + t(z_1 - z_0)[\/latex], [latex]0 \\leq t \\leq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The key difference from a full line is the restriction [latex]0 \\leq t \\leq 1[\/latex], which limits us to just the segment between the two endpoints.<\/section><section class=\"textbox example\" aria-label=\"Example\">Find parametric equations of the line segment between the points [latex]P(2, 1, 4)[\/latex] and\u00a0[latex]Q(3, -1 ,3)[\/latex].[reveal-answer q=\"112936453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"112936453\"]\r\n<p id=\"fs-id1163724069731\">By\u00a0the Parametric Equations of a Line, we have<\/p>\r\n<p style=\"text-align: center;\">[latex]x=x_0+t(x_1-x_0)[\/latex],\u00a0[latex]y=y_0+t(y_1-y_0)[\/latex],\u00a0[latex]z=z_0+t(z_1-z_0)[\/latex],\u00a0[latex]0\\leq t\\leq1[\/latex].<\/p>\r\n<p id=\"fs-id1163724054258\">Working with each component separately, we get<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\nx&amp;=x_0+t(x_1-x_0) \\\\\r\n&amp;=2+t(3-2) \\\\\r\n&amp;=2+t,\r\n\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\ny&amp;=y_0+t(y_1-y_0) \\\\\r\n&amp;=1+t(-1-1) \\\\\r\n&amp;=1-2t,\r\n\\end{aligned}[\/latex]<\/p>\r\n<p id=\"fs-id1163723280727\">and<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\nz&amp;=z_0+t(z_1-z_0) \\\\\r\n&amp;=4+t(3-4) \\\\\r\n&amp;=4-t.\r\n\\end{aligned}[\/latex]<\/p>\r\n<p id=\"fs-id1163723200089\">Therefore, the parametric equations for the line segment are<\/p>\r\n<p style=\"text-align: center;\">[latex]x=2+t,y=1-2t,z=4-t, \\ 0\\leq t\\leq1[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write different types of equations (vector, parametric, symmetric) to describe lines in space<\/li>\n<li>Find the distance from a point to a given line<\/li>\n<li>Write equations for planes using vectors and scalars<\/li>\n<li>Find the angle between two planes<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Equations for a Line in Space<\/h2>\n<p class=\"whitespace-normal break-words\">You&#8217;re already comfortable writing equations for lines in two dimensions. In three dimensions, the concept is similar, but instead of using slope to describe direction, we use a <strong>direction vector<\/strong> parallel to the line.<\/p>\n<p class=\"whitespace-normal break-words\">Two vectors are parallel when they point in the same or opposite directions. For nonzero vectors [latex]\\textbf{u}[\/latex] and [latex]\\textbf{v}[\/latex], this means there exists a scalar [latex]k[\/latex] such that [latex]\\textbf{u} = k\\textbf{v}[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>parallel vectors<\/h3>\n<p>Two nonzero vectors [latex]\\textbf{u}[\/latex] and [latex]\\textbf{v}[\/latex] are parallel if and only if [latex]\\textbf{u} = k\\textbf{v}[\/latex] for some scalar [latex]k[\/latex]. When [latex]k > 0[\/latex], the vectors have the same direction. When [latex]k < 0[\/latex], they have opposite directions.\n\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The zero vector [latex]\\textbf 0[\/latex] is considered to be parallel to all vectors.<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Vector Equation of a Line<\/h3>\n<p class=\"whitespace-normal break-words\">Consider a line [latex]L[\/latex] passing through point [latex]P(x_0, y_0, z_0)[\/latex] with direction vector [latex]\\textbf{v} = \\langle a, b, c \\rangle[\/latex]. For any point [latex]Q(x, y, z)[\/latex] on this line, the vector [latex]\\overrightarrow{PQ}[\/latex] must be parallel to [latex]\\textbf{v}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">This means [latex]\\overrightarrow{PQ} = t\\textbf{v}[\/latex] for some scalar [latex]t[\/latex]. Since [latex]\\overrightarrow{PQ} = \\langle x - x_0, y - y_0, z - z_0 \\rangle[\/latex], we have:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\langle x - x_0, y - y_0, z - z_0 \\rangle = t\\langle a, b, c \\rangle[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Using vector addition, we can rewrite this as:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\langle x, y, z \\rangle = \\langle x_0, y_0, z_0 \\rangle + t\\langle a, b, c \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Setting [latex]\\textbf{r} = \\langle x, y, z \\rangle[\/latex] and [latex]\\textbf{r}_0 = \\langle x_0, y_0, z_0 \\rangle[\/latex], we get the <strong>vector equation of a line<\/strong>:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\textbf{r} = \\textbf{r}_0 + t\\textbf{v}[\/latex]<\/p>\n<figure id=\"attachment_5157\" aria-describedby=\"caption-attachment-5157\" style=\"width: 433px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5157 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/27211934\/2.63.jpg\" alt=\"\" width=\"433\" height=\"390\" \/><figcaption id=\"caption-attachment-5157\" class=\"wp-caption-text\">Figure 1. Vector [latex]\\textbf v[\/latex] is the direction vector for [latex]\\overrightarrow{PQ}[\/latex].<\/figcaption><\/figure>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Parametric and Symmetric Equations<\/h3>\n<p class=\"whitespace-normal break-words\">From the vector equation, we can extract three component equations:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]x - x_0 = ta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y - y_0 = tb[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z - z_0 = tc[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Solving for [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex] gives us the <strong>parametric equations<\/strong>:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]x = x_0 + ta, \\quad y = y_0 + tb, \\quad z = z_0 + tc[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">If [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are all nonzero, we can solve each equation for [latex]t[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\frac{x - x_0}{a} = t, \\quad \\frac{y - y_0}{b} = t, \\quad \\frac{z - z_0}{c} = t[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Since all expressions equal [latex]t[\/latex], we can set them equal to create <strong>symmetric equations<\/strong>:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\frac{x - x_0}{a} = \\frac{y - y_0}{b} = \\frac{z - z_0}{c}[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>line equations in three dimensions<\/h3>\n<p class=\"whitespace-normal break-words\">A line [latex]L[\/latex] through point [latex]P(x_0, y_0, z_0)[\/latex] parallel to vector [latex]\\textbf{v} = \\langle a, b, c \\rangle[\/latex] can be described by:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Vector form<\/strong>: [latex]\\textbf{r} = \\textbf{r}_0 + t\\textbf{v}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Parametric form<\/strong>: [latex]x = x_0 + ta, \\quad y = y_0 + tb, \\quad z = z_0 + tc[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Symmetric form<\/strong>: [latex]\\frac{x - x_0}{a} = \\frac{y - y_0}{b} = \\frac{z - z_0}{c}[\/latex] (when [latex]a, b, c \\neq 0[\/latex])<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0The parametric and symmetric equations of a line are not unique. Using a different direction vector or different point on the line creates equivalent but different-looking equations that describe the same line.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find parametric and symmetric equations of the line passing through points [latex](1, 4, -2)[\/latex] and\u00a0[latex](-3, 5, 0)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q389475782\">Show Solution<\/button><\/p>\n<div id=\"q389475782\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163723796963\">First, identify a vector parallel to the line:<\/p>\n<p style=\"text-align: center;\">[latex]{\\bf{v}}=\\langle-3-1,5-4,0-(-2)\\rangle=\\langle-4,1,2\\rangle[\/latex].<\/p>\n<p id=\"fs-id1163724059770\">Use either of the given points on the line to complete the parametric equations:<\/p>\n<p style=\"text-align: center;\">[latex]x=1-4t[\/latex],\u00a0[latex]y=4+t[\/latex], and\u00a0[latex]z=-2+2t[\/latex].<\/p>\n<p id=\"fs-id1163723281292\">Solve each equation for [latex]t[\/latex] to create the symmetric equation of the line:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{x-1}{-4}=y-4=\\frac{z+2}2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3>Line Segments<\/h3>\n<p class=\"whitespace-normal break-words\">Sometimes you need equations for just a portion of a line, not the entire line. This requires restricting the parameter [latex]t[\/latex] to a specific interval.<\/p>\n<p class=\"whitespace-normal break-words\">Consider two points [latex]P(x_0, y_0, z_0)[\/latex] and [latex]Q(x_1, y_1, z_1)[\/latex] with position vectors [latex]\\textbf{p} = \\langle x_0, y_0, z_0 \\rangle[\/latex] and [latex]\\textbf{q} = \\langle x_1, y_1, z_1 \\rangle[\/latex]. To find the equation of the line segment between these points, we use [latex]P[\/latex] as our reference point and [latex]\\overrightarrow{PQ} = \\langle x_1 - x_0, y_1 - y_0, z_1 - z_0 \\rangle[\/latex] as our direction vector.<\/p>\n<p class=\"whitespace-normal break-words\">Starting with the vector equation [latex]\\textbf{r} = \\textbf{p} + t(\\overrightarrow{PQ})[\/latex], we can manipulate this algebraically:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} \\textbf{r} &= \\textbf{p} + t(\\overrightarrow{PQ}) \\\\ &= \\langle x_0, y_0, z_0 \\rangle + t\\langle x_1 - x_0, y_1 - y_0, z_1 - z_0 \\rangle \\\\ &= \\langle x_0, y_0, z_0 \\rangle + t(\\langle x_1, y_1, z_1 \\rangle - \\langle x_0, y_0, z_0 \\rangle) \\\\ &= \\langle x_0, y_0, z_0 \\rangle + t\\langle x_1, y_1, z_1 \\rangle - t\\langle x_0, y_0, z_0 \\rangle \\\\ &= (1-t)\\langle x_0, y_0, z_0 \\rangle + t\\langle x_1, y_1, z_1 \\rangle \\\\ &= (1-t)\\textbf{p} + t\\textbf{q} \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This gives us the <strong>vector equation for a line segment<\/strong>: [latex]\\textbf{r} = (1-t)\\textbf{p} + t\\textbf{q}[\/latex]<\/p>\n<p>We can also derive the parametric equations directly:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  \\textbf{r} &= \\textbf{p} + t(\\overrightarrow{PQ}) \\\\  \\langle x, y, z \\rangle &= \\langle x_0, y_0, z_0 \\rangle + t\\langle x_1 - x_0, y_1 - y_0, z_1 - z_0 \\rangle \\\\  &= \\langle x_0 + t(x_1 - x_0), y_0 + t(y_1 - y_0), z_0 + t(z_1 - z_0) \\rangle  \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Notice what happens at the endpoints:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">When [latex]t = 0[\/latex]: [latex]\\textbf{r} = \\textbf{p}[\/latex] (we&#8217;re at point [latex]P[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">When [latex]t = 1[\/latex]: [latex]\\textbf{r} = \\textbf{q}[\/latex] (we&#8217;re at point [latex]Q[\/latex])<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>line segment equation<\/h3>\n<p class=\"whitespace-normal break-words\">The line segment between points [latex]P[\/latex] and [latex]Q[\/latex] is described by:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Vector form<\/strong>: [latex]\\textbf{r} = (1-t)\\textbf{p} + t\\textbf{q}, \\quad 0 \\leq t \\leq 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Parametric form<\/strong>: [latex]x = x_0 + t(x_1 - x_0)[\/latex], [latex]y = y_0 + t(y_1 - y_0)[\/latex], [latex]z = z_0 + t(z_1 - z_0)[\/latex], [latex]0 \\leq t \\leq 1[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The key difference from a full line is the restriction [latex]0 \\leq t \\leq 1[\/latex], which limits us to just the segment between the two endpoints.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find parametric equations of the line segment between the points [latex]P(2, 1, 4)[\/latex] and\u00a0[latex]Q(3, -1 ,3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q112936453\">Show Solution<\/button><\/p>\n<div id=\"q112936453\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163724069731\">By\u00a0the Parametric Equations of a Line, we have<\/p>\n<p style=\"text-align: center;\">[latex]x=x_0+t(x_1-x_0)[\/latex],\u00a0[latex]y=y_0+t(y_1-y_0)[\/latex],\u00a0[latex]z=z_0+t(z_1-z_0)[\/latex],\u00a0[latex]0\\leq t\\leq1[\/latex].<\/p>\n<p id=\"fs-id1163724054258\">Working with each component separately, we get<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  x&=x_0+t(x_1-x_0) \\\\  &=2+t(3-2) \\\\  &=2+t,  \\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  y&=y_0+t(y_1-y_0) \\\\  &=1+t(-1-1) \\\\  &=1-2t,  \\end{aligned}[\/latex]<\/p>\n<p id=\"fs-id1163723280727\">and<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  z&=z_0+t(z_1-z_0) \\\\  &=4+t(3-4) \\\\  &=4-t.  \\end{aligned}[\/latex]<\/p>\n<p id=\"fs-id1163723200089\">Therefore, the parametric equations for the line segment are<\/p>\n<p style=\"text-align: center;\">[latex]x=2+t,y=1-2t,z=4-t, \\ 0\\leq t\\leq1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-530","chapter","type-chapter","status-publish","hentry"],"part":422,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/530","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/530\/revisions"}],"predecessor-version":[{"id":1670,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/530\/revisions\/1670"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/422"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/530\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=530"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=530"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=530"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=530"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}