Basic Engineering Mathematics, Fifth Edition

Vectors 267

at 45◦to the horizontal is shown in Figure 29.1. Note
that an angle of+ 45 ◦is drawn from the horizontal and
movesanticlockwise.

9N

0

a

45 

Figure 29.1

A velocity of 20m/s at− 60 ◦is shown in Figure 29.2.
Note that an angle of− 60 ◦is drawn from the horizontal
and movesclockwise.

60 

20m/s

0

b

Figure 29.2

29.3.1 Representing a vector

There are a number of ways of representing vector
quantities. These include

(a) Using bold print.

(b)

−→
AB where an arrow above two capital letters
denotes the sense of direction, where Ais the
starting point andBthe end point of the vector.
(c) ABora; i.e., a line over the top of letter.
(d) a; i.e., underlined letter.
Theforceof9Nat45◦shown in Figure 29.1 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
Figure 29.2 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.

29.4 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 and F 2 ,areshownin

Figure 29.3. 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 29.3

To add forcesF 1 andF 2 ,

(i) Force F 1 is drawn to scale horizontally,

shown as 0 ain Figure 29.4.

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

angleθto 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 29.4

(b) Parallelogram method

To add the two force vectors, F 1 and F 2 of

Figure 29.3,

(i) A linecbis constructed which is parallel to

and equal in length to 0 a(see Figure 29.5).

(ii) A lineabis constructed which is parallel to

and equal in length to 0 c.

(iii) The resultant force is given by the diagonal

of the parallelogram; i.e., length 0 b.

This procedure is called the‘parallelogram’

method.