Higher Engineering Mathematics, Sixth Edition

Vectors 253

(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.

0

c

F 1

F 2

a

b



Figure 24.5

Problem 1. A force of 5N is inclined at an angle

of 45◦to a second force of 8N, both forces acting at

a point. Find the magnitude of the resultant of these

two forces and the direction of the resultant with

respect to the 8N force by:

(a) the ‘nose-to-tail’ method, and (b) the

‘parallelogram’ method.

The two forces are shown in Fig. 24.6. (Although the
8N force is shown horizontal, it could have been drawn
in any direction.)

458

5N

8N

Figure 24.6

(a) ‘Nose-to tail’ method

(i) The 8 N force is drawn horizontally 8 units long,

shown as 0 ain Fig. 24.7

(ii) From the nose of the 8N force, the 5N force

is drawn 5 units long at an angle of 45◦to the

horizontal, shown asab

(iii) The resultant force is given by length 0band

is measured as12Nand angleθis measured

as 17 ◦.

458

5N

8N a

0

b



Figure 24.7

(b) ‘Parallelogram’ method

(i) InFig. 24.8, a lineis constructed whichis parallel

to and equal in length to the 8N force

(ii) Alineisconstructedwhichisparalleltoandequal

in length to the 5N force

(iii) The resultant force is given by the diagonal

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

measured as 12Nand angle θ is measured

as 17 ◦.

458

5N

8N

b

0



Figure 24.8

Thus,theresultantofthetwoforcevectorsinFig.24.6

is 12N at 17◦to the 8N force.

Problem 2. Forces of 15N and 10N are at an

angle of 90◦to each other as shown in Fig. 24.9.

Find, by drawing, the magnitude of the resultant of

these two forces and the direction of the resultant

with respect to the 15N force.

15N

10N

Figure 24.9

Using the ‘nose-to-tail’ method:

(i) The 15N force is drawn horizontally 15 units

long as shown in Fig. 24.10

(ii) From the nose of the 15N force, the 10N force

is drawn 10 units long at an angle of 90◦to the

horizontal as shown

(iii) Theresultant forceis shown asRand is measured

as18Nand angleθis measured as 34 ◦.