Chapter 2: Vectors in Space

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×𝐯=(𝑢2⁢𝑣3−𝑢3⁢𝑣2)⁢𝐢−(𝑢1⁢𝑣3−𝑢3⁢𝑣1)⁢𝐣+(𝑢1⁢𝑣2−𝑢2⁢𝑣1)𝐤, where 𝐮=〈𝑢1,𝑢2,𝑢3〉 and 𝐯=〈𝑣1,𝑣2,𝑣3〉

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·𝐯=𝑢1⁢𝑣1+𝑢2⁢𝑣2+𝑢3⁢𝑣3 where 𝐮=〈𝑢1,𝑢2,𝑢3〉 and 𝐯=〈𝑣1,𝑣2,𝑣3〉

ellipsoid

a three-dimensional surface where all traces of this surface are ellipses

elliptic cone

a three-dimensional surface where traces of this surface include ellipses and intersecting lines

elliptic paraboloid

a three-dimensional surface where 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=(𝑥0,𝑦0,𝑧0) is a point on the plane, and 𝑑=−𝑎⁢𝑥0−𝑏⁢𝑦0−𝑐⁢𝑧0

hyperboloid of one sheet

a three-dimensional surface where traces of this surface include ellipses and hyperbolas

hyperboloid of two sheets

a three-dimensional surface where 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=𝑥0+𝑡⁢𝑎, 𝑦=𝑦0+𝑡⁢𝑏, and 𝑧=𝑧0+𝑡⁢𝑐 describing the line with direction vector 𝐯=〈𝑎,𝑏,𝑐〉 passing through point (𝑥0,𝑦0,𝑧0)

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−𝑥0)+𝑏⁢(𝑦−𝑦0)+𝑐⁢(𝑧−𝑧0)=0 used to describe a plane containing point 𝑃=(𝑥0,𝑦0,𝑧0) with normal vector 𝐧=〈𝑎,𝑏,𝑐〉 or its alternate form 𝑎⁢𝑥+𝑏⁢𝑦+𝑐⁢𝑧+𝑑=0, where 𝑑=−𝑎⁢𝑥0−𝑏⁢𝑦0−𝑐⁢𝑧0

scalar multiplication

a vector operation that defines the product of a scalar and a vector; The product 𝑘𝐯 of a vector v and a scalar k is a vector with a magnitude that is |𝑘| times the magnitude of 𝐯, and with a direction that is the same as the direction of 𝐯 if 𝑘>0, and opposite the direction of 𝐯 if 𝑘<0.If 𝑘=0 or 𝐯=𝟎, then 𝑘⁢𝐯=𝟎.

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≠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 𝑂⁢𝑃, where 𝑂 is the origin and 0≤𝜑≤𝜋

standard equation of a sphere

(𝑥−𝑎)²+(𝑦−𝑏)²+(𝑧−𝑐)²=𝑟² 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

describes the line with direction vector 𝐯=〈𝑎,𝑏,𝑐〉 passing through point (𝑥0,𝑦0,𝑧0)

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; measures the tendency of a force to produce rotation about an axis of rotation. Let 𝐫 be a vector with an initial point located on the axis of rotation and with a terminal point located at the point where the force is applied, and let vector 𝐅 represent the force; equal to the cross product of 𝐫 and 𝐅:

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·(𝐯×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. 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 has a terminal point that coincides with the terminal point of 𝐰

vector difference

the vector difference 𝐯−𝐰 is defined as 𝐯+⁢(−𝐰)=𝐯+⁢(−1)𝐰

vector equation of a line

the equation 𝐫=𝐫0+𝑡𝐯 used to describe a line with direction vector 𝐯=〈𝑎,𝑏,𝑐〉 passing through point 𝑃=(𝑥0,𝑦0,𝑧0), where 𝐫0=〈𝑥0,𝑦0,𝑧0〉, is the position vector of point 𝑃

vector equation of a plane

the equation 𝐧·→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

zero vector

the vector with both initial point and terminal point (0,0)

Distance between two points in space

The distance 𝑑 between points (𝑥1,𝑦1,𝑧1) and (𝑥2,𝑦2,𝑧2)

Sphere with center (𝑎,𝑏,𝑐) and radius r

Dot product of u and v

given vectors 𝐮=〈𝑢1,𝑢2,𝑢3〉 and 𝐯=〈𝑣1,𝑣2,𝑣3〉; it is the sum of the products of the components

Cosine of the angle formed by 𝐮 and 𝐯

cos⁡𝜃 in terms of the dot product,

Vector projection of 𝐯 onto 𝐮

It has the same initial point as 𝐮 and 𝐯 and the same direction as 𝐮, and represents the component of 𝐯 that acts in the direction of 𝐮. If 𝜃 represents the angle between 𝐮 and 𝐯, then, by properties of triangles, we know the length of proj𝐮𝐯 is ‖proj𝐮𝐯‖ =‖𝐯‖·⁢|cos| 𝜃. When the angle 𝜃 between 𝐮 and 𝐯 is an obtuse angle, the projection will be in the opposite direction of 𝐮

Scalar projection of 𝐯 onto 𝐮

Work done by a force F to move an object through displacement vector 𝑃⁢𝑄

When a constant force is applied to an object so that the object moves in a straight line from point P to point Q, the work W done by the force F is equal to the product of the force's magnitude and the distance traveled. 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.

The cross product of two vectors in terms of the unit vectors

Let 𝐮=〈𝑢1,𝑢2,𝑢3〉⁢and𝐯=〈𝑣1,𝑣2,𝑣3〉.

Distance between a Plane and a Point

Let 𝐿 be a line in space passing through point 𝑃 with direction vector 𝐯. If 𝑀 is any point not on 𝐿