Higher Engineering Mathematics, Sixth Edition

252 Higher Engineering Mathematics

An arrow is used to denote the sense, or direction, of the

vector.

The arrow end of a vector is called the ‘nose’ and the

other end the ‘tail’.

Forexample,aforceof9Nactingat45◦tothehorizontal

is shown in Fig. 24.1.

Note that an angle of+ 45 ◦is drawn from the horizontal

and movesanticlockwise.

9N

0

a

458

Figure 24.1

A velocity of 20m/s at− 60 ◦is shown in Fig. 24.2.

Note that an angle of− 60 ◦is drawn from the horizontal

and movesclockwise.

60 

20 m/s

0

b

Figure 24.2

Representing a vector

There are a number of ways of representing vector

quantities. These include:

1. Usingbold print

2.

−→

AB where an arrow above two capital letters

denotes the sense of direction, where A is the

starting point andBthe end point of the vector

3. ABorai.e. a line over the top of letters


4. ai.e. an underlined letter
   Theforceof9Nat45◦shown in Fig. 24.1 may be
   represented as:
   0 a or

−→

0 a or 0 a

The magnitude of the force is 0a

Similarly, the velocity of 20m/s at− 60 ◦shown in

Fig. 24.2 may be represented as:

0 b or

−→

0 b or 0 b

The magnitude of the velocity is 0b

In this chapter a vector quantity is denoted bybold

print.

24.4 Addition of vectors by drawing

Adding two or more vectors by drawing assumes that

a ruler, pencil and protractor are available. Results

obtained by drawing are naturally not as accurate as

those obtained by calculation.

(a) Nose-to-tail method

Two force vectors,F 1 andF 2 , are shown in Fig. 24.3.

When an object is subjected to more than one force,

the resultant of the forces is found by the addition of

vectors.



F 2

F 1

Figure 24.3

To add forcesF 1 andF 2 :

(i) ForceF 1 is drawn to scale horizontally, shown as

0 ain Fig. 24.4.

(ii) From the nose ofF 1 ,forceF 2 is drawn at angle

θto the horizontal, shown asab.

(iii) The resultant force is given by length 0 b,which

may be measured.

This procedure is called the ‘nose-to-tail’or‘triangle’

method.



F 2

F 1 a

b

0

Figure 24.4

(b) Parallelogram method

To add the two force vectors,F 1 andF 2 , of Fig. 24.3:

(i) A linecbis constructed which is parallel to and

equal in length to 0 a(see Fig. 24.5).