Higher Engineering Mathematics, Sixth Edition

Scalar and vector products 279

The direction cosines are:

cosα=

x

√

x^2 +y^2 +z^2

=

3

√

14

=0.802

cosβ=

y

√

x^2 +y^2 +z^2

=

2

√

14

=0.535

and cosγ=

y

√

x^2 +y^2 +z^2

=

1

√

14

=0.267

(and henceα=cos−^10. 802 = 36. 7 ◦,β=cos−^10. 535 =
57.7◦andγ=cos−^10. 267 = 74. 5 ◦).
Notethatcos^2 α+cos^2 β+cos^2 γ= 0. 8022 + 0. 5352 +
0. 2672 =1.

Practical application of scalar product

Problem 6. A constant force of

F= 10 i+ 2 j−knewtons displaces an object from

A=i+j+ktoB= 2 i−j+ 3 k(in metres). Find the

work done in newton metres.

One of the applications of scalar products is to the work
done by a constant force when movinga body. The work
done is the product of the applied force and the distance
moved in the direction of the force.

i.e. work done=F•d

The principles developed in Problem 13, page 262,
apply equally to this problem when determining the
displacement. From the sketch shown in Fig. 26.8,

AB=AO+OB=OB−OA

that is AB=( 2 i−j+ 3 k)−(i+j+k)

=i− 2 j+ 2 k

A (1,1,1)

B (2, 2 1, 3)

O (0, 0, 0)

Figure 26.8

The work done isF•d,thatisF•ABin this case

i.e.work done=( 10 i+ 2 j−k)•(i− 2 j+ 2 k)

But from equation (2),

a•b=a 1 b 1 +a 2 b 2 +a 3 b 3

Hencework done=

( 10 × 1 )+( 2 ×(− 2 ))+((− 1 )× 2 )=4Nm.

(Theoretically, it is quite possible to get a negative

answer to a ‘work done’ problem. This indicates that

the force must be in the opposite sense to that given, in

order to give the displacement stated.)

Now try the following exercise

Exercise 112 Further problemson scalar

products

1. Find the scalar producta•bwhen
   (i) a=i+ 2 j−kandb= 2 i+ 3 j+k
   (ii) a=i− 3 j+kandb= 2 i+j+k
   [(i) 7 (ii) 0]
   Givenp= 2 i− 3 j,q= 4 j−kand
   r=i+ 2 j− 3 k, determine the quantities
   stated in problems 2 to 8.


2. (a)p•q (b)p•r [(a)−12 (b)−4]


3. (a)q•r (b)r•q [(a) 11 (b) 11]


4. (a)|p| (b)|r| [(a)

√

13 (b)

√

14]

5. (a)p•(q+r) (b) 2r•(q− 2 p)
   [(a)−16 (b) 38]


6. (a)|p+r| (b)|p|+|r|
   [(a)

√

19 (b) 7.347]

7. Find the angle between (a) p and q
   (b)qandr. [(a) 143.82◦(b) 44.52◦]


8. Determine the direction cosines of (a) p
   (b)q(c)r.
   ⎡
   ⎣

(a) 0. 555 ,− 0. 832 , 0

(b) 0 , 0. 970 ,− 0. 243

(c) 0. 267 , 0. 535 ,− 0. 802

⎤

⎦

9. Determine the angle between the forces:

F 1 = 3 i+ 4 j+ 5 kand

F 2 =i+j+k [11. 54 ◦]