Vocabulary and Review Problems

Algebraic properties of vectors

The following properties hold for vectors , and in and scalars and in .

(a)
Commutative Property of Addition

(b)
Associative Property of Addition

(c)
Existence of Additive Identity:

There exists a vector such that

(d)
Existence of Additive Inverse:

For every vector , there exists a vector such that

(e)
Distributive Property over Vector Addition

(f)
Distributive Property over Scalar Addition

(g)
Associative Property for Scalar Multiplication

(h)
Multiplication by 1

Angle between vectors

Let and be vectors in , and let be the included angle. Then

Cross product

Let and be vectors in . The cross product of and , denoted by , is given by

Direction vector for a line

A vector parallel to the line.

Distance between two points

Let and be points in . The distance between and is given by

Dot product

Let and be vectors in . The dot product of and , denoted by , is given by

Dot product properties

The following properties hold for vectors , and in and scalar in .

(a)
(b)
(c)
(d)
(e)
, and if and only if .
(f)

Equation of a plane

The plane through with a normal vector is given by

Head-to-tail rule for vector addition

Length (norm, magnitude) of a vector

Let be a vector in , then the length, or the magnitude, of is given by

Normal vector

A vector perpendicular to a plane is said to be a normal vector to the plane.

Orthogonal vectors

Let and be vectors in . We say and are orthogonal if .

Parallel vectors

Two non-zero vectors are parallel if they are non-zero scalar multiples of each other.

Parallelogram rule for vector addition

Parametric equations of a line

Let be a direction vector for line in , and let be an arbitrary point on . Then the following parametric equations describe :

Projection of a vector onto a vector

Let be a vector, and let be a non-zero vector. The projection of onto is given by

Pythagoras’ Theorem

You know the one! Where have we used it in the context of points? vectors?

Scalar multiplication

Let be a vector in , and let be a scalar, then

Standard unit vectors

Let denote a vector that has as the component and zeros elsewhere. In other words, where is in the position. We say that is a standard unit vector of .

Unit vector

A vector of length 1.

Vector

A quantity possessing magnitude and direction.

Vector addition

Let and be vectors in . We define by

Zero vector

A vector of zero length. The zero vector has no direction.