Key Terms

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

$\text{u}\times \text{v}=\left(u_2{v}_3-u_3{v}_2\right)\text{i}-\left(u_1{v}_3-u_3{v}_1\right)\text{j}+\left(u_1{v}_2-u_2{v}_1\right)\text{k},$ 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

$\text{u}·\text{v}=u_1{v}_1+u_2{v}_2+u_3{v}_3$ 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 $d=\text{−}a{x}_0-b{y}_0-c{z}_0$

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 $d=\text{−}a{x}_0-b{y}_0-c{z}_0$

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 $\stackrel{—}{OP},$ 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: $\text{u}·\left(\text{v}\times \text{w}\right)$

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 $v+\left(\text{−}\text{w}\right)=v+(\mathrm{-1})\text{w}$

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 $\text{n}·\overrightarrow{PQ}=0,$ 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 $\left(0,0\right)$