Chapter 26

Scalar and vector products

26.1 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 10m/s at 50◦is

10m/sat50◦

10m/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.
Onemethod of completely specifyingthedirectionof
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. 26.1. Such a system is called a
unit triad.

y

x

o

z

k

j

i

Figure 26.1

In Fig. 26.2, 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. 26.3,opbeing
shown in Fig. 26.3(a) andorin Fig. 26.3(b).

y

x

z

k

j

a b

r

iO

Figure 26.2

(a)

(b)

k

P

j

i

4

3

22

O

i

r j

k

O

5

2

22

Figure 26.3