### Key Terms

- component
- a scalar that describes either the vertical or horizontal direction of a vector

- coordinate plane
- a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the
*xy*-plane,*xz*-plane, or the*yz*-plane

- cross product
- $\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}=\left({u}_{2}{v}_{3}-{u}_{3}{v}_{2}\right)\text{i}-\left({u}_{1}{v}_{3}-{u}_{3}{v}_{1}\right)\text{j}+\left({u}_{1}{v}_{2}-{u}_{2}{v}_{1}\right)\text{k},$ where $\text{u}=\langle {u}_{1},{u}_{2},{u}_{3}\rangle $ and $\text{v}=\langle {v}_{1},{v}_{2},{v}_{3}\rangle $

- 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 $\left(r,\theta ,z\right),$ where $\left(r,\theta \right)$ represents the polar coordinates of the point’s projection in the
*xy*-plane, and $z$ represents the point’s projection onto the*z*-axis

- determinant
- a real number associated with a square matrix

- direction angles
- the angles formed by a nonzero vector and the coordinate axes

- direction cosines
- the cosines of the angles formed by a nonzero vector and the coordinate axes

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

- dot product or scalar product
- $\text{u}\xb7\text{v}={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+{u}_{3}{v}_{3}$ where $\text{u}=\langle {u}_{1},{u}_{2},{u}_{3}\rangle $ and $\text{v}=\langle {v}_{1},{v}_{2},{v}_{3}\rangle $

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

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

- elliptic paraboloid
- a three-dimensional surface described by an equation of the form $z=\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}};$ traces of this surface include ellipses and parabolas

- equivalent vectors
- vectors that have the same magnitude and the same direction

- general form of the equation of a plane
- an equation in the form $ax+by+cz+d=0,$ where $\text{n}=\langle a,b,c\rangle $ is a normal vector of the plane, $P=\left({x}_{0},{y}_{0},{z}_{0}\right)$ is a point on the plane, and $d=\text{\u2212}a{x}_{0}-b{y}_{0}-c{z}_{0}$

- hyperboloid of one sheet
- a three-dimensional surface described by an equation of the form $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-\frac{{z}^{2}}{{c}^{2}}=1;$ traces of this surface include ellipses and hyperbolas

- hyperboloid of two sheets
- a three-dimensional surface described by an equation of the form $\frac{{z}^{2}}{{c}^{2}}-\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1;$ traces of this surface include ellipses and hyperbolas

- initial point
- the starting point of a vector

- magnitude
- the length of a vector

- normal vector
- a vector perpendicular to a plane

- normalization
- using scalar multiplication to find a unit vector with a given direction

- octants
- the eight regions of space created by the coordinate planes

- orthogonal vectors
- vectors that form a right angle when placed in standard position

- parallelepiped
- a three-dimensional prism with six faces that are parallelograms

- parallelogram method
- a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram

- parametric equations of a line
- the set of equations $x={x}_{0}+ta,$ $y={y}_{0}+tb,$ and $z={z}_{0}+tc$ describing the line with direction vector $\text{v}=\langle a,b,c\rangle $ passing through point $\left({x}_{0},{y}_{0},{z}_{0}\right)$

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

- right-hand rule
- a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the
*z*-axis in such a way that the fingers curl from the positive*x*-axis to the positive*y*-axis, the thumb points in the direction of the positive*z*-axis

- rulings
- parallel lines that make up a cylindrical surface

- scalar
- a real number

- scalar equation of a plane
- the equation $a\left(x-{x}_{0}\right)+b\left(y-{y}_{0}\right)+c\left(z-{z}_{0}\right)=0$ used to describe a plane containing point $P=\left({x}_{0},{y}_{0},{z}_{0}\right)$ with normal vector $\text{n}=\langle a,b,c\rangle $ or its alternate form $ax+by+cz+d=0,$ where $d=\text{\u2212}a{x}_{0}-b{y}_{0}-c{z}_{0}$

- scalar multiplication
- a vector operation that defines the product of a scalar and a vector

- scalar projection
- the magnitude of the vector projection of a vector

- skew lines
- two lines that are not parallel but do not intersect

- sphere
- the set of all points equidistant from a given point known as the
*center*

- spherical coordinate system
- a way to describe a location in space with an ordered triple $\left(\rho ,\theta ,\phi \right),$ where $\rho $ is the distance between $P$ and the origin $\left(\rho \ne 0\right),$ $\theta $ is the same angle used to describe the location in cylindrical coordinates, and $\phi $ is the angle formed by the positive
*z*-axis and line segment $\stackrel{\u2014}{OP},$ where $O$ is the origin and $0\le \phi \le \pi $

- standard equation of a sphere
- ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}+{\left(z-c\right)}^{2}={r}^{2}$ describes a sphere with center $\left(a,b,c\right)$ and radius $r$

- standard unit vectors
- unit vectors along the coordinate axes: $\text{i}=\langle 1,0\rangle ,\text{j}=\langle 0,1\rangle $

- standard-position vector
- a vector with initial point $\left(0,0\right)$

- symmetric equations of $\text{a}$ line
- the equations $\frac{x-{x}_{0}}{a}=\frac{y-{y}_{0}}{b}=\frac{z-{z}_{0}}{c}$ describing the line with direction vector $\text{v}=\langle a,b,c\rangle $ passing through point $\left({x}_{0},{y}_{0},{z}_{0}\right)$

- terminal point
- the endpoint of a vector

- three-dimensional rectangular coordinate system
- a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple $\left(x,y,z\right)$ that plots its location relative to the defining axes

- torque
- the effect of a force that causes an object to rotate

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

- triangle inequality
- the length of any side of a triangle is less than the sum of the lengths of the other two sides

- triangle method
- a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector

- triple scalar product
- the dot product of a vector with the cross product of two other vectors: $\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right)$

- unit vector
- a vector with margnitude $1$

- vector
- a mathematical object that has both magnitude and direction

- vector addition
- a vector operation that defines the sum of two vectors

- vector difference
- the vector difference $\text{v}-\text{w}$ is defined as $v+\left(\text{\u2212}\text{w}\right)=v+(\mathrm{-1})\text{w}$

- vector equation of a line
- the equation $\text{r}={\text{r}}_{0}+t\text{v}$ used to describe a line with direction vector $\text{v}=\langle a,b,c\rangle $ passing through point $P=\left({x}_{0},{y}_{0},{z}_{0}\right),$ where ${\text{r}}_{0}=\langle {x}_{0},{y}_{0},{z}_{0}\rangle ,$ is the position vector of point $P$

- vector equation of a plane
- the equation $\text{n}\xb7\overrightarrow{PQ}=0,$ where $P$ is a given point in the plane, $Q$ is any point in the plane, and $\text{n}$ is a normal vector of the plane

- vector product
- the cross product of two vectors

- vector projection
- the component of a vector that follows a given direction

- vector sum
- the sum of two vectors, $\text{v}$ and $\text{w},$ can be constructed graphically by placing the initial point of $\text{w}$ at the terminal point of $\text{v};$ then the vector sum $v+w$ is the vector with an initial point that coincides with the initial point of $\text{v},$ and with a terminal point that coincides with the terminal point of $\text{w}$

- work done by a force
- work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector
**F**and the displacement of an object by a vector**s**, then the work done by the force is the dot product of**F**and**s**.

- zero vector
- the vector with both initial point and terminal point $\left(0,0\right)$