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0593_C02_fm Page 15 Monday, May 6, 2002 1:46 PM

2

Review of Vector Algebra

2.1 Introduction
In Chapter 1, we reviewed the basic concepts of vectors. We considered vectors, scalars,
and the multiplication of vectors and scalars. We also examined zero vectors and unit
vectors. In this chapter, we will build upon these ideas as we develop a review of vector
algebra. Specifically, we will review the concepts of vector equality, vector addition, and
vector multiplication. We will also review the concepts of reference frames and unit vector
sets. Finally, we will review the elementary procedures of matrix algebra.

2.2 Equality of Vectors, Fixed and Free Vectors

Recall that the characteristics of a vector are its magnitude, its orientation, and its sense.
Indeed, we could say that a vector is defined by its characteristics. The concept of vector
equality follows from this definition: Specifically, two vectors are equal if (and only if)
they have the same characteristics. Vector equality is fundamental to the development of
vector algebra. For example, if vectors are equal, they may be interchanged in vector
equations, which enables us to simplify expressions. It should be noted, however, that
vector equality does not necessarily denote physical equality, particularly when the vec-
tors model physical quantities. This occurs, for example, with forces. We will explore this
concept later.
Two fundamental ideas useful in relating mathematical and physical quantities are the
concepts of fixed and free vectors. A fixed vector has its location restricted to a line fixed
in space. To illustrate this, consider the fixed line L as shown in Figure 2.2.1. Let v be a
vector whose location is restricted to L, and let the location of v along L be arbitrary. Then
v is a fixed vector. Because the location of v along L is arbitrary, v might even be called a
sliding vector.
Alternatively, a free vector is a vector that may be placed anywhere in space if its
characteristics are maintained. Unit vectors such as n , n , and n shown in Figure 2.2.1
y
z
x
are examples of free vectors. Most vectors in our analyses will be free vectors. Indeed, we
will assume that vectors are free vectors unless otherwise stated.

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