Essential Concepts

Cross Product and Applications

• 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 [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex] is a vector orthogonal to both [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex]. Its length is given by [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 [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex]. Its direction is given by the right-hand rule.
• The algebraic formula for calculating the cross product of two vectors, [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex], is [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].
• The cross product satisfies the following properties for vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex], and scalar [latex]c[/latex]:
  •  [latex]{\bf{u}}\times{\bf{v}}=-({\bf{v}}\times{\bf{u}})[/latex]
  • [latex]{\bf{u}}\times\left({\bf{v}}+{\bf{w}}\right)={\bf{u}}\times{\bf{v}}+{\bf{u}}\times{\bf{w}}[/latex]
  • [latex]c({\bf{u}}\times{\bf{v}})=(c{\bf{u}})\times{\bf{v}}={\bf{u}}\times(c{\bf{v}})[/latex]
  • [latex]{\bf{u}}\times{0}={0}\times{\bf{u}}=0[/latex]
  • [latex]{\bf{v}}\times{\bf{v}}=0[/latex]
  • [latex]{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})=({\bf{u}}\times{\bf{v}})\cdot{\bf{w}}[/latex]
• The cross product of vectors  [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [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]
• If vectors [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] form adjacent sides of a parallelogram, then the area of the parallelogram is given by [latex]\parallel{\bf{u}}\times{\bf{v}}\parallel[/latex].
• The triple scalar product of vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex] is [latex]{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})[/latex].
• The volume of a parallelepiped with adjacent edges given by vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex] is [latex]V=|{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})|[/latex]
• If the triple scalar product of vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex] is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
• The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
• Torque [latex]\tau[/latex] measures the tendency of a force to produce rotation about an axis of rotation. If force [latex]{\bf{F}}[/latex] is acting at a distance [latex]{\bf{r}}[/latex] from the axis, then torque is equal to the cross product of [latex]{\bf{r}}[/latex] and [latex]{\bf{F}}[/latex]: [latex]\tau={\bf{r}}\times{\bf{F}}[/latex].

Lines and Planes in Space

• 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] passing through point [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] is the position vector of point [latex]P[/latex] This equation can be rewritten to form the parametric equations of the line: [latex]x=x_{0}+ta[/latex], [latex]y=y_{0}+tb[/latex], and [latex]z=z_{0}+tc[/latex].  The 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].
• Let [latex]L[/latex] be a line in space passing through point [latex]P[/latex] with direction vector [latex]{\bf{v}}[/latex]. If [latex]Q[/latex] is any point not on [latex]L[/latex] then the distance from [latex]Q[/latex] to [latex]L[/latex] is [latex]d=\frac{\parallel\overrightarrow{PQ}\times{\bf{v}}\parallel}{\parallel{\bf{v}}\parallel}[/latex].
• In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
• Given a point [latex]P[/latex] and vector [latex]{\bf{n}}[/latex] the set of all points [latex]Q[/latex] satisfying equation [latex]{\bf{n}}\cdot\overrightarrow{PQ}=0[/latex] forms a plane. Equation [latex]{\bf{n}}\cdot\overrightarrow{PQ}=0[/latex] is known as the vector equation of a plane.
• The scalar equation of a plane containing point [latex]P=(x_{0},y_{0},z_{0})[/latex] with normal vector [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 [latex]ax+by+cz+d=0[/latex], where [latex]d=-ax_{0}-by_{0}-cz_{0}[/latex]. This form of the equation is sometimes called the general form of the equation of a plane.
• Suppose a plane with normal vector [latex]{\bf{n}}[/latex] passes through point [latex]Q[/latex]. The distance [latex]D[/latex] from the plane to point [latex]P[/latex] not in the plane is given by [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].
• The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
• 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 [latex]{\bf{n}}_{1}[/latex] and [latex]{\bf{n}}_{2}[/latex] are normal vectors to the planes.
• The distance [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]

Alternative Coordinate Systems and Surfaces

• 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’s projection in the [latex]xy[/latex]-plane and [latex]z[/latex] represents the point’s projection onto the [latex]z[/latex]-axis.
• 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].
• 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].
• 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] is the distance between [latex]P[/latex] and the origin [latex](\rho\ne{0})[/latex], [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], where [latex]O[/latex] is the origin and [latex]0\le\varphi\le\pi[/latex].
• To convert a point from spherical coordinates to Cartesian coordinates, use equations [latex]x=\rho\sin\varphi\cos\theta[/latex], [latex]y=\rho\sin\varphi\sin\theta[/latex], and [latex]z=\rho\cos\varphi[/latex].
• 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]
• 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].
• 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]
• A set of lines parallel to a given line passing through a given curve is called a cylinder, or a cylindrical surface. The parallel lines are called rulings.
• The intersection of a three-dimensional surface and a plane is called a trace. 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.
• 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].
• To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.

Key Equations

• The cross product of two vectors in terms of the unit vectors
  [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]
• Vector Equation of a Line
  [latex]{\bf{r}}={\bf{r}}_{0}+t{\bf{v}}[/latex]
• Parametric Equations of a Line
  [latex]\dfrac{x-x_{0}}{a}=\dfrac{y-y_{0}}{b}=\dfrac{z-z_{0}}{c}[/latex]
• Vector Equation of a Plane
  [latex]{\bf{n}}\cdot\overrightarrow{PQ}=0[/latex]
• Scalar Equation of a Plane
  [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[/latex]
• Distance between a Plane and a Point
  [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]

Glossary

cross product

[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 [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex]

cylinder

a set of lines parallel to a given line passing through a given curve

cylindrical coordinate system

a way to describe a location in space with an ordered triple [latex](r,\theta,z)[/latex], where [latex](r,\theta)[/latex] represents the polar coordinates of the point’s projection in the [latex]xy[/latex]-plane, and [latex]z[/latex] represents the point’s projection onto the [latex]z[/latex]-axis

determinant

a real number associated with a square matrix

direction vector

a vector parallel to a line that is used to describe the direction, or orientation, of the line in space

ellipsoid

a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/latex] all traces of this surface are ellipses

elliptic cone

a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0[/latex] traces of this surface include ellipses and intersecting lines

elliptic paraboloid

a three-dimensional surface described by an equation of the form [latex]z=\frac{x^2}{a^2}+\frac{y^2}{b^2}[/latex] traces of this surface include ellipses and parabolas

general form of the equation of a plane

an equation in the form [latex]ax+by+cz+d=0[/latex], where [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] is a point on the plane, and [latex]d=-ax_{0}-by_{0}-cz_{0}[/latex]

hyperboloid of one sheet

a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1[/latex] traces of this surface include ellipses and parabolas

hyperboloid of two sheets

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] traces of this surface include ellipses and parabolas

normal vector

a vector perpendicular to a plane

parallelpiped

a three-dimensional prism with six faces that are parallelograms

parametric equations of a line:

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 [latex]{\bf{v}}=\langle{a,b,c}\rangle[/latex] passing through point [latex](x_{0},y_{0},z_{0})[/latex]

quadric surfaces

surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)

rulings

parallel lines that make up a cylindrical surface

scalar equation of a plane:

the equation [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[/latex] used to describe a plane containing point [latex]P=(x_{0},y_{0},z_{0})[/latex] with normal vector [latex]{\bf{n}}=\langle{a,b,c}\rangle[/latex] or its alternate form [latex]ax+by+cz+d=0[/latex], where [latex]d=-ax_{0}-by_{0}-cz_{0}[/latex]

skew lines:

two lines that are not parallel but do not intersect

spherical coordinate system

a way to describe a location in space with an ordered triple [latex](\rho,\theta,\varphi)[/latex], where [latex]\rho[/latex] is the distance between [latex]P[/latex] and the origin [latex]\rho \ne {0}[/latex], [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] where [latex]O[/latex] is the origin and [latex]0\le\varphi\le\pi[/latex]

symmetric equations of a line:

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 [latex]{\bf{v}}=\langle{a,b,c}\rangle[/latex] passing through point [latex](x_{0},y_{0},z_{0})[/latex]

torque

the effect of a force that causes an object to rotate

trace

the intersection of a three-dimensional surface with a coordinate plane

triple scalar product

the dot product of a vector with the cross product of two other vectors: [latex]{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})[/latex]

vector equation of a line:

the equation [latex]{\bf{r}} ={\bf{r}}_{0}+t{\bf{v}}[/latex] used 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], where [latex]{\bf{r}}_{0}=\langle{x_{0},y_{0},z_{0}}\rangle[/latex] is the position vector of point [latex]P[/latex]

vector equation of a plane:

the equation [latex]{\bf{n}}\cdot\overrightarrow{PQ}=0[/latex],
where [latex]P[/latex] is a given point in the plane, [latex]Q[/latex] is any point in the plane, and [latex]{\bf{n}}[/latex] is a normal vector of the plane

vector product

the cross product of two vectors