{"id":48,"date":"2021-07-30T17:06:04","date_gmt":"2021-07-30T17:06:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=48"},"modified":"2022-10-21T00:25:23","modified_gmt":"2022-10-21T00:25:23","slug":"summary-of-equations-of-lines-and-planes-in-space","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-equations-of-lines-and-planes-in-space\/","title":{"raw":"Summary of Equations of Lines and Planes in Space","rendered":"Summary of Equations of Lines and Planes in Space"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector [latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex] is [latex]{\\bf{r}}={\\bf{r}}_{0}+t{\\bf{v}}[\/latex], where [latex]{\\bf{r}}_{0}=\\langle{x_{0},y_{0},z_{0}}\\rangle[\/latex]\u00a0is the position vector of point\u00a0[latex]P[\/latex]\u00a0This equation can be rewritten to form the parametric equations of the line: [latex]x=x_{0}+ta[\/latex],\u00a0[latex]y=y_{0}+tb[\/latex], and\u00a0[latex]z=z_{0}+tc[\/latex].\u00a0\u00a0The line can also be described with the symmetric equations [latex]\\frac{x-x_{0}}{a}=\\frac{y-y_{0}}{b}=\\frac{z-z_{0}}{c}[\/latex].<\/li>\r\n \t<li>Let [latex]L[\/latex]\u00a0be a line in space passing through point [latex]P[\/latex]\u00a0with direction vector\u00a0[latex]{\\bf{v}}[\/latex]. If\u00a0[latex]Q[\/latex]\u00a0is any point not on\u00a0[latex]L[\/latex]\u00a0then the distance from\u00a0[latex]Q[\/latex] to\u00a0[latex]L[\/latex] is [latex]d=\\frac{\\parallel\\overrightarrow{PQ}\\times{\\bf{v}}\\parallel}{\\parallel{\\bf{v}}\\parallel}[\/latex].<\/li>\r\n \t<li>In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.<\/li>\r\n \t<li>Given a point\u00a0[latex]P[\/latex]\u00a0and vector\u00a0[latex]{\\bf{n}}[\/latex]\u00a0the set of all points [latex]Q[\/latex]\u00a0satisfying equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex] forms a plane. Equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex] is\u00a0known as the\u00a0<em data-effect=\"italics\">vector equation of a plane.<\/em><\/li>\r\n \t<li>The scalar equation of a plane containing point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0with normal vector\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex] is [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]. This equation can be expressed as\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex].\u00a0This form of the equation is sometimes called the\u00a0<em data-effect=\"italics\">general form of the equation of a plane<\/em>.<\/li>\r\n \t<li>Suppose a plane with normal vector [latex]{\\bf{n}}[\/latex]\u00a0passes through point\u00a0[latex]Q[\/latex].\u00a0The distance\u00a0[latex]D[\/latex]\u00a0from the plane to point\u00a0[latex]P[\/latex]\u00a0not in the plane is given by\u00a0[latex]D=\\parallel\\text{proj}_{\\bf{n}}\\overrightarrow{QP}\\parallel=|\\text{comp}_{\\bf{n}}\\overrightarrow{QP}|=\\dfrac{|\\overrightarrow{QP}\\cdot{\\bf{n}}|}{\\parallel{\\bf{n}}\\parallel}[\/latex].<\/li>\r\n \t<li>The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.<\/li>\r\n \t<li>The measure of the angle [latex]\\theta[\/latex] between two intersecting planes can be found using the equation: [latex]\\cos\\theta=\\frac{|{\\bf{n}}_{1}\\cdot{\\bf{n}}_{2}|}{\\parallel{\\bf{n}}_{1}\\parallel\\parallel{\\bf{n}}_{2}\\parallel}[\/latex], where\u00a0[latex]{\\bf{n}}_{1}[\/latex] and [latex]{\\bf{n}}_{2}[\/latex] are normal vectors to the planes.<\/li>\r\n \t<li>The distance\u00a0[latex]D[\/latex] from the point [latex](x_{0},y_{0},z_{0})[\/latex] to plane [latex]ax+by+cz+d=0[\/latex] is given by [latex]D=\\dfrac{|a(x_{0}-x_{1})+b(y_{0}-y_{1})+c(z_{0}-z_{1})|}{\\sqrt{a^2+b^2+c^2}}=\\dfrac{|ax_{0}+by_{0}+cz_{0}+d|}{\\sqrt{a^2+b^2+c^2}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Vector Equation of a Line<\/strong>\r\n[latex]{\\bf{r}}={\\bf{r}}_{0}+t{\\bf{v}}[\/latex]<\/li>\r\n \t<li><strong>Parametric Equations of a Line<\/strong>\r\n[latex]\\dfrac{x-x_{0}}{a}=\\dfrac{y-y_{0}}{b}=\\dfrac{z-z_{0}}{c}[\/latex]<\/li>\r\n \t<li><strong>Vector Equation of a Plane<\/strong>\r\n[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex]<\/li>\r\n \t<li><strong>Scalar Equation of a Plane<\/strong>\r\n[latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Distance between a Plane and a Point<\/strong>\r\n[latex]D=\\parallel\\text{proj}_{\\bf{n}}\\overrightarrow{QP}\\parallel=|\\text{comp}_{\\bf{n}}\\overrightarrow{QP}|=\\dfrac{|\\overrightarrow{QP}\\cdot{\\bf{n}}|}{\\parallel{\\bf{n}}\\parallel}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>direction vector<\/dt>\r\n \t<dd>a vector parallel to a line that is used to describe the direction, or orientation, of the line in space<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>general form of the equation of a plane<\/dt>\r\n \t<dd>an equation in the form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex] is a normal vector of the plane, [latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0is a point on the plane, and\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>normal vector<\/dt>\r\n \t<dd>a vector perpendicular to a plane<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>parametric equations of a line:<\/dt>\r\n \t<dd>the set of equations [latex]x=x_{0}+ta[\/latex], [latex]y=y_{0}+tb[\/latex], and [latex]z=z_{0}+tc[\/latex] describing the line with direction vector\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar equation of a plane:<\/dt>\r\n \t<dd>the equation [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]\u00a0used to describe a plane containing point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0with normal vector\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0or its alternate form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>skew lines:<\/dt>\r\n \t<dd>two lines that are not parallel but do not intersect<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>symmetric equations of a line:<\/dt>\r\n \t<dd>the equations [latex]\\frac{x-x_{0}}{a}=\\frac{y-y_{0}}{b}=\\frac{z-z_{0}}{c}[\/latex] describing the line with direction vector\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\r\n \t<dd><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector equation of a line:<\/dt>\r\n \t<dd>the equation\u00a0[latex]{\\bf{r}} ={\\bf{r}}_{0}+t{\\bf{v}}[\/latex]\u00a0used to describe a line with direction vector [latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex] passing through point [latex]P=(x_{0},y_{0},z_{0})[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex]{\\bf{r}}_{0}=\\langle{x_{0},y_{0},z_{0}}\\rangle[\/latex]\u00a0is the position vector of point [latex]P[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector equation of a plane:<\/dt>\r\n \t<dd>the equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex],\r\nwhere [latex]P[\/latex]\u00a0is a given point in the plane, [latex]Q[\/latex] is any point in the plane, and\u00a0[latex]{\\bf{n}}[\/latex] is a normal vector of the plane<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector [latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex] is [latex]{\\bf{r}}={\\bf{r}}_{0}+t{\\bf{v}}[\/latex], where [latex]{\\bf{r}}_{0}=\\langle{x_{0},y_{0},z_{0}}\\rangle[\/latex]\u00a0is the position vector of point\u00a0[latex]P[\/latex]\u00a0This equation can be rewritten to form the parametric equations of the line: [latex]x=x_{0}+ta[\/latex],\u00a0[latex]y=y_{0}+tb[\/latex], and\u00a0[latex]z=z_{0}+tc[\/latex].\u00a0\u00a0The line can also be described with the symmetric equations [latex]\\frac{x-x_{0}}{a}=\\frac{y-y_{0}}{b}=\\frac{z-z_{0}}{c}[\/latex].<\/li>\n<li>Let [latex]L[\/latex]\u00a0be a line in space passing through point [latex]P[\/latex]\u00a0with direction vector\u00a0[latex]{\\bf{v}}[\/latex]. If\u00a0[latex]Q[\/latex]\u00a0is any point not on\u00a0[latex]L[\/latex]\u00a0then the distance from\u00a0[latex]Q[\/latex] to\u00a0[latex]L[\/latex] is [latex]d=\\frac{\\parallel\\overrightarrow{PQ}\\times{\\bf{v}}\\parallel}{\\parallel{\\bf{v}}\\parallel}[\/latex].<\/li>\n<li>In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.<\/li>\n<li>Given a point\u00a0[latex]P[\/latex]\u00a0and vector\u00a0[latex]{\\bf{n}}[\/latex]\u00a0the set of all points [latex]Q[\/latex]\u00a0satisfying equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex] forms a plane. Equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex] is\u00a0known as the\u00a0<em data-effect=\"italics\">vector equation of a plane.<\/em><\/li>\n<li>The scalar equation of a plane containing point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0with normal vector\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex] is [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]. This equation can be expressed as\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex].\u00a0This form of the equation is sometimes called the\u00a0<em data-effect=\"italics\">general form of the equation of a plane<\/em>.<\/li>\n<li>Suppose a plane with normal vector [latex]{\\bf{n}}[\/latex]\u00a0passes through point\u00a0[latex]Q[\/latex].\u00a0The distance\u00a0[latex]D[\/latex]\u00a0from the plane to point\u00a0[latex]P[\/latex]\u00a0not in the plane is given by\u00a0[latex]D=\\parallel\\text{proj}_{\\bf{n}}\\overrightarrow{QP}\\parallel=|\\text{comp}_{\\bf{n}}\\overrightarrow{QP}|=\\dfrac{|\\overrightarrow{QP}\\cdot{\\bf{n}}|}{\\parallel{\\bf{n}}\\parallel}[\/latex].<\/li>\n<li>The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.<\/li>\n<li>The measure of the angle [latex]\\theta[\/latex] between two intersecting planes can be found using the equation: [latex]\\cos\\theta=\\frac{|{\\bf{n}}_{1}\\cdot{\\bf{n}}_{2}|}{\\parallel{\\bf{n}}_{1}\\parallel\\parallel{\\bf{n}}_{2}\\parallel}[\/latex], where\u00a0[latex]{\\bf{n}}_{1}[\/latex] and [latex]{\\bf{n}}_{2}[\/latex] are normal vectors to the planes.<\/li>\n<li>The distance\u00a0[latex]D[\/latex] from the point [latex](x_{0},y_{0},z_{0})[\/latex] to plane [latex]ax+by+cz+d=0[\/latex] is given by [latex]D=\\dfrac{|a(x_{0}-x_{1})+b(y_{0}-y_{1})+c(z_{0}-z_{1})|}{\\sqrt{a^2+b^2+c^2}}=\\dfrac{|ax_{0}+by_{0}+cz_{0}+d|}{\\sqrt{a^2+b^2+c^2}}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Vector Equation of a Line<\/strong><br \/>\n[latex]{\\bf{r}}={\\bf{r}}_{0}+t{\\bf{v}}[\/latex]<\/li>\n<li><strong>Parametric Equations of a Line<\/strong><br \/>\n[latex]\\dfrac{x-x_{0}}{a}=\\dfrac{y-y_{0}}{b}=\\dfrac{z-z_{0}}{c}[\/latex]<\/li>\n<li><strong>Vector Equation of a Plane<\/strong><br \/>\n[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex]<\/li>\n<li><strong>Scalar Equation of a Plane<\/strong><br \/>\n[latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Distance between a Plane and a Point<\/strong><br \/>\n[latex]D=\\parallel\\text{proj}_{\\bf{n}}\\overrightarrow{QP}\\parallel=|\\text{comp}_{\\bf{n}}\\overrightarrow{QP}|=\\dfrac{|\\overrightarrow{QP}\\cdot{\\bf{n}}|}{\\parallel{\\bf{n}}\\parallel}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>direction vector<\/dt>\n<dd>a vector parallel to a line that is used to describe the direction, or orientation, of the line in space<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>general form of the equation of a plane<\/dt>\n<dd>an equation in the form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex] is a normal vector of the plane, [latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0is a point on the plane, and\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>normal vector<\/dt>\n<dd>a vector perpendicular to a plane<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>parametric equations of a line:<\/dt>\n<dd>the set of equations [latex]x=x_{0}+ta[\/latex], [latex]y=y_{0}+tb[\/latex], and [latex]z=z_{0}+tc[\/latex] describing the line with direction vector\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar equation of a plane:<\/dt>\n<dd>the equation [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]\u00a0used to describe a plane containing point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0with normal vector\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0or its alternate form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>skew lines:<\/dt>\n<dd>two lines that are not parallel but do not intersect<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>symmetric equations of a line:<\/dt>\n<dd>the equations [latex]\\frac{x-x_{0}}{a}=\\frac{y-y_{0}}{b}=\\frac{z-z_{0}}{c}[\/latex] describing the line with direction vector\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\n<dd><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector equation of a line:<\/dt>\n<dd>the equation\u00a0[latex]{\\bf{r}} ={\\bf{r}}_{0}+t{\\bf{v}}[\/latex]\u00a0used to describe a line with direction vector [latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex] passing through point [latex]P=(x_{0},y_{0},z_{0})[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex]{\\bf{r}}_{0}=\\langle{x_{0},y_{0},z_{0}}\\rangle[\/latex]\u00a0is the position vector of point [latex]P[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector equation of a plane:<\/dt>\n<dd>the equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex],<br \/>\nwhere [latex]P[\/latex]\u00a0is a given point in the plane, [latex]Q[\/latex] is any point in the plane, and\u00a0[latex]{\\bf{n}}[\/latex] is a normal vector of the plane<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-48\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) 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