Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR ALGEBRA

i

j

k

axi

ayj

azk

a

Figure 7.7 A Cartesian basis set. The vectorais the sum ofaxi,ayjandazk.

basis vectorsi,jandkmay themselves be represented by (1, 0 ,0), (0, 1 ,0) and

(0, 0 ,1) respectively.

We can consider the addition and subtraction of vectors in terms of their

components. The sum of two vectorsaandbis found by simply adding their

components, i.e.

a+b=axi+ayj+azk+bxi+byj+bzk

=(ax+bx)i+(ay+by)j+(az+bz)k, (7.11)

and their difference by subtracting them,

a−b=axi+ayj+azk−(bxi+byj+bzk)

=(ax−bx)i+(ay−by)j+(az−bz)k. (7.12)

Two particles have velocitiesv 1 =i+3j+6kandv 2 =i− 2 k, respectively. Find the

velocityuof the second particle relative to the first.

The required relative velocity is given by

u=v 2 −v 1 =(1−1)i+(0−3)j+(− 2 −6)k

=− 3 j− 8 k.

7.5 Magnitude of a vector

The magnitude of the vectorais denoted by|a|ora. In terms of its components

in three-dimensional Cartesian coordinates, the magnitude ofais given by

a≡|a|=

√

a^2 x+a^2 y+a^2 z. (7.13)

Hence, the magnitude of a vector is a measure of its length. Such an analogy is

useful for displacement vectors but magnitude is better described, for example, by

‘strength’ for vectors such as force or by ‘speed’ for velocity vectors. For instance,