Higher Engineering Mathematics

SCALAR AND VECTOR PRODUCTS 241

D

Note that cos^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−k newtons displaces an object

from A=i+j+k to B= 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 moving a 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 8, Chapter 21,
apply equally to this problem when determining the
displacement. From the sketch shown in Fig. 22.8,

AB=AO+OB=OB−OA

that is AB=(2i−j+ 3 k)−(i+j+k)

=i− 2 j+ 2 k

O (0, 0 , 0)

A (1, 1 , 1)

B (2, − 1 , 3)

Figure 22.8

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

i.e.work done=(10i+ 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 97 Further problems on 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)pandq(b)q
   andr [(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 ◦]

10. Find the angle between the velocity vectors
    υ 1 = 5 i+ 2 j+ 7 kandυ 2 = 4 i+j−k

[66.40◦]

11. Calculate the work done by a force
    F=(− 5 i+j+ 7 k) N when its point of appli-
    cation moves from point (− 2 i− 6 j+k)m
    to the point (i−j+ 10 k) m. [53 Nm]

22.3 Vector products

A second product of two vectors is called thevec-

tororcross productand is defined in terms of its

modulus and the magnitudes of the two vectors and