Higher Engineering Mathematics

D

Vector geometry

22

Scalar and vector products

22.1 The unit triad

When a vectorxof magnitudexunits and directionθ◦
is divided by the magnitude of the vector, the result
is a vector of unit length at angleθ◦. The unit vector

for a velocity of 10 m/s at 50◦is

10 m/sat50◦

10 m/s

, i.e.

1at50◦. In general, the unit vector foroais

oa

|oa|

,

theoabeing a vector and having both magnitude
and direction and|oa|being the magnitude of the
vector only.
One method of completely specifying the direc-
tion of a vector in space relative to some reference
point is to use three unit vectors, mutually at right
angles to each other, as shown in Fig. 22.1. Such a
system is called aunit triad.

z

x

o j y

i

k

Figure 22.1

In Fig. 22.2, one way to get fromotoris to movex
units alongito pointa, thenyunits in directionjto
get toband finallyzunits in directionkto get tor.
The vectororis specified as

or=xi+yj+zk

Problem 1. With reference to three axes drawn

mutually at right angles, depict the vectors

(i)op= 4 i+ 3 j− 2 kand (ii)or= 5 i− 2 j+ 2 k.

The required vectors are depicted in Fig. 22.3,op
being shown in Fig. 22.3(a) andorin Fig. 22.3(b).

y

a

i

O

x j

z

k

r

b

Figure 22.2

k

i

4

3

O j

− 2

P

(a)

k

j

i

5

2

(b)

− 2

O

r

Figure 22.3