Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.4 BASIS VECTORS AND COMPONENTS

These equations are consistent and have the solutionλ=μ=2/3. Substituting these
values into either (7.7) or (7.8) we find that the position vector of the centroidGis given
by

g=^13 (a+b+c).

7.4 Basis vectors and components

Given any three different vectorse 1 ,e 2 ande 3 , which do not all lie in a plane,

it is possible, in a three-dimensional space, to write any other vector in terms of

scalar multiples of them:

a=a 1 e 1 +a 2 e 2 +a 3 e 3. (7.9)

The three vectorse 1 ,e 2 ande 3 are said to form abasis(for the three-dimensional

space); the scalarsa 1 ,a 2 anda 3 , which may be positive, negative or zero, are

called thecomponentsof the vectorawith respect to this basis. We say that the

vector has beenresolvedinto components.

Most often we shall use basis vectors that are mutually perpendicular, for ease

of manipulation, though this is not necessary. In general, a basis set must

(i) have as many basis vectors as the number of dimensions (in more formal

language, the basis vectors must span the space) and

(ii) be such that no basis vector may be described as a sum of the others, or,

more formally, the basis vectors must belinearly independent. Putting this

mathematically, inNdimensions, we require

c 1 e 1 +c 2 e 2 +···+cNeN= 0 ,

for any set of coefficientsc 1 ,c 2 ,...,cNexceptc 1 =c 2 =···=cN=0.

In this chapter we will only consider vectors in three dimensions; higher dimen-

sionality can be achieved by simple extension.

If we wish to label points in space using a Cartesian coordinate system (x, y, z),

we may introduce the unit vectorsi,jandk, which point along the positivex-,

y-andz- axes respectively. A vectoramay then be written as a sum of three

vectors, each parallel to a different coordinate axis:

a=axi+ayj+azk. (7.10)

A vector in three-dimensional space thus requires three components to describe

fully both its direction and its magnitude. A displacement in space may be

thought of as the sum of displacements along thex-,y-andz- directions (see

figure 7.7). For brevity, the components of a vectorawith respect to a particular

coordinate system are sometimes written in the form (ax,ay,az). Note that the