Understanding Engineering Mathematics

Multiplication by a positive (negative)λleaves the direction ofaunchanged (reversed).
−ais defined by(− 1 )a.
Vectorsa=a 1 i+a 2 j+a 3 k,b=b 1 i+b 2 j+b 3 kare added and subtracted ‘componen-
twise’:
a+b=(a 1 +b 1 )i+(a 2 +b 2 )j+(a 3 +b 3 )k
a−b=(a 1 −b 1 )i+(a 2 −b 2 )j+(a 3 −b 3 )k

It is obvious from this that, as noted in Section 11.3,

a+b=b+a (vector addition is commutative)

a+(b+c)=(a+b)+c (vector addition is associative)

Problem 11.6

For a=i−j, b=2iY3jY4k, c=−i−2j−4k, find the vectors repre-

senting

(i) 3a (ii) aYb (iii) aY2b−c(iv)2a−3c

Using the rules given above we have

(i) 3 a= 3 (i−j)= 3 i− 3 j

(ii) a+b=i−j+ 2 i+ 3 j+ 4 k= 3 i+ 2 j+ 4 k

(iii) a+ 2 b−c=i−j+ 2 ( 2 i+ 3 j+ 4 k)−(−i− 2 j− 4 k)

=i−j+ 4 i+ 6 j+ 8 k+i+ 2 j+ 4 k

= 6 i+ 7 j+ 12 k

(iv) 2 a− 3 c= 2 (i−j)− 3 (−i− 2 j− 4 k)

= 2 i− 2 j+ 3 i+ 6 j+ 12 k

= 5 i+ 4 j+ 12 k

Exercises on 11.9

1. Ifa=μi+ 2 j+(λ−μ)k,b= 2 λi+υj+ 2 k
   and
   2 a+b= 0

determineμ,λ,υ.

2. Referring to Q1, evaluate

(i) |a| (ii) aˆ (iii) 3b (iv) a+ 2 k (v) 3a−b

Answers

1.μ=

1

2

,λ=−

1

2

,υ=− 4