Key Terms - Calculus Volume 3

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: $u \times v = (u_{2} v_{3} - u_{3} v_{2}) i - (u_{1} v_{3} - u_{3} v_{1}) j + (u_{1} v_{2} - u_{2} v_{1}) k,$ where $u = 〈 u_{1}, u_{2}, u_{3} 〉$ and $v = 〈 v_{1}, v_{2}, v_{3} 〉$

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 $(r, θ, z),$ where $(r, θ)$ 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: $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}$ where $u = 〈 u_{1}, u_{2}, u_{3} 〉$ and $v = 〈 v_{1}, v_{2}, v_{3} 〉$

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 $a x + b y + c z + d = 0,$ where $n = 〈 a, b, c 〉$ is a normal vector of the plane, $P = (x_{0}, y_{0}, z_{0})$ is a point on the plane, and $d = − a x_{0} - b y_{0} - c z_{0}$

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 $x = x_{0} + t a,$ $y = y_{0} + t b,$ and $z = z_{0} + t c$ describing the line with direction vector $v = 〈 a, b, c 〉$ passing through point $(x_{0}, y_{0}, z_{0})$

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 $a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0$ used to describe a plane containing point $P = (x_{0}, y_{0}, z_{0})$ with normal vector $n = 〈 a, b, c 〉$ or its alternate form $a x + b y + c z + d = 0,$ where $d = − a x_{0} - b y_{0} - c z_{0}$

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 $(ρ, θ, φ),$ where $ρ$ is the distance between $P$ and the origin $(ρ \neq 0),$ $θ$ is the same angle used to describe the location in cylindrical coordinates, and $φ$ is the angle formed by the positive z-axis and line segment $\overset{—}{O P},$ where $O$ is the origin and $0 \leq φ \leq π$

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

standard unit vectors: unit vectors along the coordinate axes: $i = 〈 1, 0 〉, j = 〈 0, 1 〉$

standard-position vector: a vector with initial point $(0, 0)$

symmetric equations of $a$ line: the equations $\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}$ describing the line with direction vector $v = 〈 a, b, c 〉$ passing through point $(x_{0}, y_{0}, z_{0})$

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 $(x, y, z)$ 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: $u \cdot (v \times w)$

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 $v - w$ is defined as $v + (− w) = v + (−1) w$

vector equation of a line: the equation $r = r_{0} + t v$ used to describe a line with direction vector $v = 〈 a, b, c 〉$ passing through point $P = (x_{0}, y_{0}, z_{0}),$ where $r_{0} = 〈 x_{0}, y_{0}, z_{0} 〉,$ is the position vector of point $P$

vector equation of a plane: the equation $n \cdot \vec{P Q} = 0,$ where $P$ is a given point in the plane, $Q$ is any point in the plane, and $n$ 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, $v$ and $w,$ can be constructed graphically by placing the initial point of $w$ at the terminal point of $v;$ then the vector sum $v + w$ is the vector with an initial point that coincides with the initial point of $v,$ and with a terminal point that coincides with the terminal point of $w$

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 $(0, 0)$