Understanding Engineering Mathematics

Exercises on 11.8

1. Express the position vectors with the given endpoints in terms ofi,j,kvectors.

(i) (3,−1, 2) (ii) (1, 0, 1) (iii) (a,b,c) (iv) (− 1 , 1 ,− 1 )

2. If(α−β)i+(β+ 2 χ)j+(α−χ)k= 2 i−kdetermineα,β,χ.

Answers

1. (i) 3i−j+ 2 k (ii)i+k (iii)ai+bj+ck (iv)−i+j−k

2.α=0,β=−2,χ= 1

11.9 Properties of vectors

Two vectorsa=a 1 i+a 2 j+a 3 kb=b 1 i+b 2 j+b 3 k

referred to the same axes areequalif and only if

a 1 =b 1 ,a 2 =b 2 ,a 3 =b 3

(ai=bi for alli)

Azero vectoris one whose components are all zero:

0 = 0 i+ 0 j+ 0 k

LetA,Bbe points with coordinates(a 1 ,a 2 ,a 3 ),(b 1 ,b 2 ,b 3 )with respect toOxyzaxes.
Theposition vector ofBrelative toAis (see Figure 11.11)

AB=

−→

AB=(b 1 −a 1 )i+(b 2 −a 2 )j+(b 3 −a 3 )k

x

O

y

z

A(a 1 ,a 2 ,a 3 )

AB B(b^1 ,b^2 ,b^3 )

Figure 11.11Position vector ofBrelative toA.

As a particular case letA=( 0 , 0 , 0 ), the origin, andB= a pointP(x,y,z), as shown
in Figure 11.12.
−→
OP or OPorr=xi+yj+zkis called theposition vectorofP (with respect to the
axesOxyz). We also refer toras aradius vector.