Higher Engineering Mathematics, Sixth Edition

254 Higher Engineering Mathematics

Thus,the resultant of the two force vectors is 18N at

34 ◦to the 15N force.

10N

R

15N



Figure 24.10

Problem 3. Velocities of 10m/s, 20m/s and

15m/s act as shown in Fig. 24.11. Determine, by

drawing, the magnitude of the resultant velocity and

its direction relative to the horizontal.

158

(^3)
(^2)
(^1)
308
10m/s
20m/s
15m/s
Figure 24.11
When more than two vectors are being added the ‘nose-
to-tail’ method is used.
The order in which the vectors are added does not
matter. In this case the order taken isv 1 ,thenv 2 ,then
v 3. However, if a different order is taken the same result
will occur.
(i) v 1 is drawn 10 units long at an angle of 30◦to the
horizontal, shown as 0 ain Fig. 24.12
(ii) From the nose ofv 1 ,v 2 is drawn 20 units long at
an angle of 90◦to the horizontal, shown asab
(iii) From the nose ofv 2 ,v 3 is drawn 15 units long at
an angle of 195◦to the horizontal, shown asbr
(iv) The resultant velocity is given by length 0 rand
is measured as22m/sand the angle measured to
the horizontal is 105 ◦.
Thus,the resultant of the three velocities is 22m/s at
105 ◦to the horizontal.
b
195 
105 
30 
O
r
a
Figure 24.12
Worked Problems 1 to 3 have demonstrated how
vectors are added to determine their resultant and their
direction. However, drawing to scale is time-consuming
and not highly accurate. The following sections demon-
strate how to determine resultant vectors by calculation
using horizontal and vertical components and, where
possible, by Pythagoras’s theorem.

24.5 Resolving vectors into horizontal

and vertical components

A force vectorFis shown in Fig. 24.13 at angleθto the

horizontal. Such a vector can be resolved into two com-

ponents such that the vector addition of the components

is equal to the original vector.



F

Figure 24.13

The two components usually taken are ahorizontal

componentand avertical component.

If a right-angled triangle is constructed as shown in

Fig. 24.14, then 0ais called the horizontal component

ofFandabis called the vertical component ofF.

(^0) a
F
b

Figure 24.14