{"id":516,"date":"2025-10-09T15:17:46","date_gmt":"2025-10-09T15:17:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=516"},"modified":"2025-10-14T16:25:24","modified_gmt":"2025-10-14T16:25:24","slug":"advanced-vector-operations-and-applications-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/advanced-vector-operations-and-applications-cheat-sheet\/","title":{"raw":"Advanced Vector Operations and Applications: Cheat Sheet","rendered":"Advanced Vector Operations and Applications: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Cross Product and Applications<\/strong>\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<strong>Lines and Planes in Space<\/strong>\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<strong>Alternative Coordinate Systems and Surfaces<\/strong>\r\n<ul>\r\n \t<li>In the cylindrical coordinate system, a point in space is represented by the ordered triple [latex](r,\\theta,z)[\/latex], where [latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane and [latex]z[\/latex]\u00a0represents the point\u2019s projection onto the [latex]z[\/latex]-axis.<\/li>\r\n \t<li>To convert a point from cylindrical coordinates to Cartesian coordinates, use equations [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]z=z[\/latex].<\/li>\r\n \t<li>To convert a point from Cartesian coordinates to cylindrical coordinates, use equations [latex]r^2=x^2+y^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]z=z[\/latex].<\/li>\r\n \t<li>In the spherical coordinate system, a point [latex]P[\/latex] in space is represented by the ordered triple [latex](\\rho,\\theta,\\varphi)[\/latex], where [latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex] and the origin [latex](\\rho\\ne{0})[\/latex], [latex]\\theta[\/latex] is the same\u00a0angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the\u00a0angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex], where [latex]O[\/latex] is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex].<\/li>\r\n \t<li>To convert a point from spherical coordinates to Cartesian coordinates, use equations\u00a0[latex]x=\\rho\\sin\\varphi\\cos\\theta[\/latex], [latex]y=\\rho\\sin\\varphi\\sin\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\r\n \t<li>To convert a point from Cartesian coordinates to spherical coordinates, use equations [latex]\\rho^{2}=x^2+y^2+z^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{x^2+y^2+z^2}}\\right)[\/latex]<\/li>\r\n \t<li>To convert a point from spherical coordinates to cylindrical coordinates, use equations [latex]r=\\rho\\sin\\varphi[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\r\n \t<li>To convert a point from cylindrical coordinates to spherical coordinates, use equations [latex]\\rho=\\sqrt{r^2+z^2}[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{r^2+z^2}}\\right)[\/latex]<\/li>\r\n \t<li>A set of lines parallel to a given line passing through a given curve is called a\u00a0<em data-effect=\"italics\">cylinder<\/em>, or a\u00a0<em data-effect=\"italics\">cylindrical surface<\/em>. The parallel lines are called\u00a0<em data-effect=\"italics\">rulings<\/em>.<\/li>\r\n \t<li>The intersection of a three-dimensional surface and a plane is called a\u00a0<em data-effect=\"italics\">trace<\/em>. To find the trace in the [latex]xy[\/latex]-, [latex]yz[\/latex]-, or [latex]xz[\/latex]-planes, set [latex]z=0[\/latex], [latex]x=0[\/latex], or [latex]y=0[\/latex], respectively.<\/li>\r\n \t<li>Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form [latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/li>\r\n \t<li>To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul>\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 \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 \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<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>cylinder<\/dt>\r\n \t<dd>a set of lines parallel to a given line passing through a given curve<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>cylindrical coordinate system<\/dt>\r\n \t<dd>a way to describe a location in space with an ordered triple [latex](r,\\theta,z)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane, and [latex]z[\/latex] represents the point's projection onto the [latex]z[\/latex]-axis<\/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>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>ellipsoid<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]\u00a0all traces of this surface are ellipses<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>elliptic cone<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=0[\/latex]\u00a0traces of this surface include ellipses and intersecting lines<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>elliptic paraboloid<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]z=\\frac{x^2}{a^2}+\\frac{y^2}{b^2}[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/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>hyperboloid of one sheet<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>hyperboloid of two sheets<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form [latex]\\frac{z^2}{c^2}-\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/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>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>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>quadric surfaces<\/dt>\r\n \t<dd>surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>rulings<\/dt>\r\n \t<dd>parallel lines that make up a cylindrical surface<\/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>spherical coordinate system<\/dt>\r\n \t<dd>a way to describe a location in space with an ordered triple\u00a0[latex](\\rho,\\theta,\\varphi)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex]\u00a0and the origin [latex]\\rho \\ne\r\n{0}[\/latex],\u00a0[latex]\\theta[\/latex] is the same angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex]<strong>\u00a0<\/strong>where\u00a0[latex]O[\/latex]\u00a0is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex]<\/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<\/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>trace<\/dt>\r\n \t<dd>the intersection of a three-dimensional surface with a coordinate plane<\/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 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>\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":"<h2>Essential Concepts<\/h2>\n<p><strong>Cross Product and Applications<\/strong><\/p>\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<p><strong>Lines and Planes in Space<\/strong><\/p>\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<p><strong>Alternative Coordinate Systems and Surfaces<\/strong><\/p>\n<ul>\n<li>In the cylindrical coordinate system, a point in space is represented by the ordered triple [latex](r,\\theta,z)[\/latex], where [latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane and [latex]z[\/latex]\u00a0represents the point\u2019s projection onto the [latex]z[\/latex]-axis.<\/li>\n<li>To convert a point from cylindrical coordinates to Cartesian coordinates, use equations [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]z=z[\/latex].<\/li>\n<li>To convert a point from Cartesian coordinates to cylindrical coordinates, use equations [latex]r^2=x^2+y^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]z=z[\/latex].<\/li>\n<li>In the spherical coordinate system, a point [latex]P[\/latex] in space is represented by the ordered triple [latex](\\rho,\\theta,\\varphi)[\/latex], where [latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex] and the origin [latex](\\rho\\ne{0})[\/latex], [latex]\\theta[\/latex] is the same\u00a0angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the\u00a0angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex], where [latex]O[\/latex] is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex].<\/li>\n<li>To convert a point from spherical coordinates to Cartesian coordinates, use equations\u00a0[latex]x=\\rho\\sin\\varphi\\cos\\theta[\/latex], [latex]y=\\rho\\sin\\varphi\\sin\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\n<li>To convert a point from Cartesian coordinates to spherical coordinates, use equations [latex]\\rho^{2}=x^2+y^2+z^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{x^2+y^2+z^2}}\\right)[\/latex]<\/li>\n<li>To convert a point from spherical coordinates to cylindrical coordinates, use equations [latex]r=\\rho\\sin\\varphi[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\n<li>To convert a point from cylindrical coordinates to spherical coordinates, use equations [latex]\\rho=\\sqrt{r^2+z^2}[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{r^2+z^2}}\\right)[\/latex]<\/li>\n<li>A set of lines parallel to a given line passing through a given curve is called a\u00a0<em data-effect=\"italics\">cylinder<\/em>, or a\u00a0<em data-effect=\"italics\">cylindrical surface<\/em>. The parallel lines are called\u00a0<em data-effect=\"italics\">rulings<\/em>.<\/li>\n<li>The intersection of a three-dimensional surface and a plane is called a\u00a0<em data-effect=\"italics\">trace<\/em>. To find the trace in the [latex]xy[\/latex]-, [latex]yz[\/latex]-, or [latex]xz[\/latex]-planes, set [latex]z=0[\/latex], [latex]x=0[\/latex], or [latex]y=0[\/latex], respectively.<\/li>\n<li>Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form [latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/li>\n<li>To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul>\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<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<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<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>cylinder<\/dt>\n<dd>a set of lines parallel to a given line passing through a given curve<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>cylindrical coordinate system<\/dt>\n<dd>a way to describe a location in space with an ordered triple [latex](r,\\theta,z)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane, and [latex]z[\/latex] represents the point&#8217;s projection onto the [latex]z[\/latex]-axis<\/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>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>ellipsoid<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]\u00a0all traces of this surface are ellipses<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>elliptic cone<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=0[\/latex]\u00a0traces of this surface include ellipses and intersecting lines<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>elliptic paraboloid<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]z=\\frac{x^2}{a^2}+\\frac{y^2}{b^2}[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/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>hyperboloid of one sheet<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>hyperboloid of two sheets<\/dt>\n<dd>a three-dimensional surface described by an equation of the form [latex]\\frac{z^2}{c^2}-\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/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>parallelpiped<\/dt>\n<dd>a three-dimensional prism with six faces that are parallelograms<\/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>quadric surfaces<\/dt>\n<dd>surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>rulings<\/dt>\n<dd>parallel lines that make up a cylindrical surface<\/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>spherical coordinate system<\/dt>\n<dd>a way to describe a location in space with an ordered triple\u00a0[latex](\\rho,\\theta,\\varphi)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex]\u00a0and the origin [latex]\\rho \\ne  {0}[\/latex],\u00a0[latex]\\theta[\/latex] is the same angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex]<strong>\u00a0<\/strong>where\u00a0[latex]O[\/latex]\u00a0is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex]<\/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<\/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>trace<\/dt>\n<dd>the intersection of a three-dimensional surface with a coordinate plane<\/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 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<dl class=\"definition\">\n<dt>vector product<\/dt>\n<dd>the cross product of two vectors<\/dd>\n<\/dl>\n","protected":false},"author":15,"menu_order":1,"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-516","chapter","type-chapter","status-publish","hentry"],"part":422,"module-header":"cheat_sheet","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\/516","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":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/516\/revisions"}],"predecessor-version":[{"id":942,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/516\/revisions\/942"}],"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\/516\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=516"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=516"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=516"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=516"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}