Basic Engineering Mathematics, Fifth Edition

Vectors 269

(iii) The resultant force is shown asRand is measured

as18Nand angleθis measured as 34 ◦.

Thus,the resultant of the two force vectors is 18N at
34 ◦to the 15N force.

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

15m/s act as shown in Figure 29.11. Determine, by

drawing, the magnitude of the resultant velocity and

its direction relative to the horizontal

15 

 3

 2

 1

30 

10m/s

20m/s

15m/s

Figure 29.11

When more than 2 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 isν 1 ,
thenν 2 ,thenν 3. However, if a different order is taken
the same result will occur.

(i) ν 1 is drawn 10 units long at an angle of 30◦to the

horizontal, shown as 0 ain Figure 29.12.

b

195 

105 

0

30 

 1

 3  2

r

a

Figure 29.12

(ii) From the nose ofν 1 ,ν 2 is drawn 20 units long at

an angle of 90◦to the horizontal, shown asab.

(iii) From the nose ofν 2 ,ν 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.

Worked Examples 1 to 3 have demonstrated how vec-

tors 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’ theorem.

29.5 Resolving vectors into horizontal

and vertical components

A force vectorFis shown in Figure 29.13 at angleθ

to the horizontal. Such a vector can be resolved into

two components such that the vector addition of the

components is equal to the original vector.



F

Figure 29.13

The two components usually taken are ahorizontal

componentand avertical component.Ifaright-angled

triangle is constructed as shown in Figure 29.14, 0ais

called the horizontal component ofFandabis called

the vertical component ofF.

(^0) a
F
b

Figure 29.14
From trigonometry(see Chapter 21 and remember SOH
CAH TOA),
cosθ=
0 a
0 b
,from which 0a= 0 bcosθ=Fcosθ