Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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,

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