{"id":46,"date":"2021-07-30T17:06:02","date_gmt":"2021-07-30T17:06:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=46"},"modified":"2022-10-21T00:18:42","modified_gmt":"2022-10-21T00:18:42","slug":"summary-of-the-cross-product","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-the-cross-product\/","title":{"raw":"Summary of the Cross Product","rendered":"Summary of the Cross Product"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The cross product [latex]{\\bf{u}}\\times{\\bf{v}}[\/latex] of two vectors [latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]\u00a0is a vector orthogonal to both\u00a0[latex]{\\bf{u}}[\/latex] and\u00a0[latex]{\\bf{v}}[\/latex].\u00a0Its length is given by\u00a0[latex]\\parallel{\\bf{u}}\\times{\\bf{v}}\\parallel=\\parallel{\\bf{u}}\\parallel\\cdot\\parallel{\\bf{v}}\\parallel\\cdot\\sin\\theta[\/latex], where [latex]\\theta[\/latex] is the angle between\u00a0[latex]{\\bf{u}}[\/latex] and\u00a0[latex]{\\bf{v}}[\/latex].\u00a0Its direction is given by the right-hand rule.<\/li>\r\n \t<li>The algebraic formula for calculating the cross product of two vectors,\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex], is\u00a0[latex]{\\bf{u}}\\times{\\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\\bf{k}}[\/latex].<\/li>\r\n \t<li>The cross product satisfies the following properties for vectors [latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex], and scalar [latex]c[\/latex]:\r\n<ul>\r\n \t<li>\u00a0[latex]{\\bf{u}}\\times{\\bf{v}}=-({\\bf{v}}\\times{\\bf{u}})[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{u}}\\times\\left({\\bf{v}}+{\\bf{w}}\\right)={\\bf{u}}\\times{\\bf{v}}+{\\bf{u}}\\times{\\bf{w}}[\/latex]<\/li>\r\n \t<li>[latex]c({\\bf{u}}\\times{\\bf{v}})=(c{\\bf{u}})\\times{\\bf{v}}={\\bf{u}}\\times(c{\\bf{v}})[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{u}}\\times{0}={0}\\times{\\bf{u}}=0[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{v}}\\times{\\bf{v}}=0[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})=({\\bf{u}}\\times{\\bf{v}})\\cdot{\\bf{w}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The cross product of vectors\u00a0\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex] is the determinant [latex]\\left|\\begin{array}{ccc} {\\bf{i}} &amp; {\\bf{j}} &amp; {\\bf{k}} \\\\ u_{1} &amp; u_{2} &amp; u_{3} \\\\ v_{1} &amp; v_{2} &amp; v_{3}\\end{array}\\right|[\/latex]<\/li>\r\n \t<li>If vectors [latex]{\\bf{u}}[\/latex] and\u00a0[latex]{\\bf{v}}[\/latex]\u00a0form adjacent sides of a parallelogram, then the area of the parallelogram is given by [latex]\\parallel{\\bf{u}}\\times{\\bf{v}}\\parallel[\/latex].<\/li>\r\n \t<li>The triple scalar product of vectors\u00a0[latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex] is\u00a0[latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})[\/latex].<\/li>\r\n \t<li>The volume of a parallelepiped with adjacent edges given by vectors\u00a0[latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex] is\u00a0[latex]V=|{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})|[\/latex]<\/li>\r\n \t<li>If the triple scalar product of vectors\u00a0[latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex] is\u00a0zero,\u00a0then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.<\/li>\r\n \t<li>The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.<\/li>\r\n \t<li>Torque [latex]\\tau[\/latex]\u00a0measures the tendency of a force to produce rotation about an axis of rotation. If force\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is acting at a distance\u00a0[latex]{\\bf{r}}[\/latex]\u00a0from the axis, then torque is equal to the cross product of\u00a0[latex]{\\bf{r}}[\/latex] and\u00a0[latex]{\\bf{F}}[\/latex]:\u00a0[latex]\\tau={\\bf{r}}\\times{\\bf{F}}[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>The cross product of two vectors in terms of the unit vectors<\/strong>\r\n[latex]{\\bf{u}}\\times{\\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\\bf{k}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>cross product<\/dt>\r\n \t<dd>[latex]{\\bf{u}}\\times{\\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\\bf{k}}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>determinant<\/dt>\r\n \t<dd>a real number associated with a square matrix<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>parallelpiped<\/dt>\r\n \t<dd>a three-dimensional prism with six faces that are parallelograms<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>torque<\/dt>\r\n \t<dd>the effect of a force that causes an object to rotate<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triple scalar product<\/dt>\r\n \t<dd>the dot product of a vector with the cross product of two other vectors: [latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector product<\/dt>\r\n \t<dd>the cross product of two vectors<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The cross product [latex]{\\bf{u}}\\times{\\bf{v}}[\/latex] of two vectors [latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]\u00a0is a vector orthogonal to both\u00a0[latex]{\\bf{u}}[\/latex] and\u00a0[latex]{\\bf{v}}[\/latex].\u00a0Its length is given by\u00a0[latex]\\parallel{\\bf{u}}\\times{\\bf{v}}\\parallel=\\parallel{\\bf{u}}\\parallel\\cdot\\parallel{\\bf{v}}\\parallel\\cdot\\sin\\theta[\/latex], where [latex]\\theta[\/latex] is the angle between\u00a0[latex]{\\bf{u}}[\/latex] and\u00a0[latex]{\\bf{v}}[\/latex].\u00a0Its direction is given by the right-hand rule.<\/li>\n<li>The algebraic formula for calculating the cross product of two vectors,\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex], is\u00a0[latex]{\\bf{u}}\\times{\\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\\bf{k}}[\/latex].<\/li>\n<li>The cross product satisfies the following properties for vectors [latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex], and scalar [latex]c[\/latex]:\n<ul>\n<li>\u00a0[latex]{\\bf{u}}\\times{\\bf{v}}=-({\\bf{v}}\\times{\\bf{u}})[\/latex]<\/li>\n<li>[latex]{\\bf{u}}\\times\\left({\\bf{v}}+{\\bf{w}}\\right)={\\bf{u}}\\times{\\bf{v}}+{\\bf{u}}\\times{\\bf{w}}[\/latex]<\/li>\n<li>[latex]c({\\bf{u}}\\times{\\bf{v}})=(c{\\bf{u}})\\times{\\bf{v}}={\\bf{u}}\\times(c{\\bf{v}})[\/latex]<\/li>\n<li>[latex]{\\bf{u}}\\times{0}={0}\\times{\\bf{u}}=0[\/latex]<\/li>\n<li>[latex]{\\bf{v}}\\times{\\bf{v}}=0[\/latex]<\/li>\n<li>[latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})=({\\bf{u}}\\times{\\bf{v}})\\cdot{\\bf{w}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The cross product of vectors\u00a0\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex] is the determinant [latex]\\left|\\begin{array}{ccc} {\\bf{i}} & {\\bf{j}} & {\\bf{k}} \\\\ u_{1} & u_{2} & u_{3} \\\\ v_{1} & v_{2} & v_{3}\\end{array}\\right|[\/latex]<\/li>\n<li>If vectors [latex]{\\bf{u}}[\/latex] and\u00a0[latex]{\\bf{v}}[\/latex]\u00a0form adjacent sides of a parallelogram, then the area of the parallelogram is given by [latex]\\parallel{\\bf{u}}\\times{\\bf{v}}\\parallel[\/latex].<\/li>\n<li>The triple scalar product of vectors\u00a0[latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex] is\u00a0[latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})[\/latex].<\/li>\n<li>The volume of a parallelepiped with adjacent edges given by vectors\u00a0[latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex] is\u00a0[latex]V=|{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})|[\/latex]<\/li>\n<li>If the triple scalar product of vectors\u00a0[latex]{\\bf{u}}[\/latex],\u00a0[latex]{\\bf{v}}[\/latex], and\u00a0[latex]{\\bf{w}}[\/latex] is\u00a0zero,\u00a0then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.<\/li>\n<li>The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.<\/li>\n<li>Torque [latex]\\tau[\/latex]\u00a0measures the tendency of a force to produce rotation about an axis of rotation. If force\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is acting at a distance\u00a0[latex]{\\bf{r}}[\/latex]\u00a0from the axis, then torque is equal to the cross product of\u00a0[latex]{\\bf{r}}[\/latex] and\u00a0[latex]{\\bf{F}}[\/latex]:\u00a0[latex]\\tau={\\bf{r}}\\times{\\bf{F}}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>The cross product of two vectors in terms of the unit vectors<\/strong><br \/>\n[latex]{\\bf{u}}\\times{\\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\\bf{k}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>cross product<\/dt>\n<dd>[latex]{\\bf{u}}\\times{\\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\\bf{k}}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>determinant<\/dt>\n<dd>a real number associated with a square matrix<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>parallelpiped<\/dt>\n<dd>a three-dimensional prism with six faces that are parallelograms<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>torque<\/dt>\n<dd>the effect of a force that causes an object to rotate<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triple scalar product<\/dt>\n<dd>the dot product of a vector with the cross product of two other vectors: [latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector product<\/dt>\n<dd>the cross product of two vectors<\/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-46\">\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":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) 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