Higher Engineering Mathematics

VECTORS, PHASORS AND THE COMBINATION OF WAVEFORMS 227

D

Often it is easier to use the‘nose-to-tail’ method
when more than two vectors are being added. The
order in which the vectors are added is immaterial.
In this case the order taken isv 1 , thenv 2 , thenv 3 but
just the same result would have been obtained if the
order had been, say,v 1 ,v 3 and finallyv 2 .v 1 is drawn
10 units long at an angle of 20◦to the horizontal,
shown byoain Fig. 21.6.v 2 is added tov 1 by drawing
a line 15 units long vertically upwards froma, shown
asab. Finally,v 3 is added tov 1 +v 2 by drawing a
line 7 units long at an angle at 190◦fromb, shown
asbr. The resultant of vector addition isorand by
measurement is 17.5 units long at an angle of 82◦to
the horizontal.

Figure 21.6

Thus

v 1 +v 2 +v 3 = 17 .5m/sat82◦to the horizontal

21.3 Resolution of vectors

A vector can be resolved into two component parts
such that the vector addition of the component parts
is equal to the original vector. The two compo-
nents usually taken are a horizontal component and
a vertical component. For the vector shown asFin
Fig. 21.7, the horizontal component isFcosθand
the vertical component isFsinθ.

Figure 21.7

For the vectorsF 1 andF 2 shown in Fig. 21.8, the

horizontal component of vector addition is:

H=F 1 cosθ 1 +F 2 cosθ 2

and the vertical component of vector addition is:

V=F 1 sinθ 1 +F 2 sinθ 2

Figure 21.8

Having obtainedHandV, the magnitude of the

resultant vectorRis given by

√

(H^2 +V^2 ) and its

angle to the horizontal is given bytan−^1 (V/H).

Problem 3. Resolve the acceleration vector of

17 m/s^2 at an angle of 120◦to the horizontal into

a horizontal and a vertical component.

For a vectorAat angleθto the horizontal, the hori-

zontal component is given byAcosθand the vertical

component byAsinθ. Any convention of signs may

be adopted, in this case horizontally from left to right

is taken as positive and vertically upwards is taken

as positive.

Horizontal component H=17 cos 120◦=−8.5

m/s^2 , acting from left to right Vertical compo-

nentV=17 sin 120◦= 14 .72 m/s^2 , acting vertically

upwards. These component vectors are shown in

Fig. 21.9.

Problem 4. Calculate the resultant force of the

two forces given in Problem 1.

With reference to Fig. 21.4(a):

Horizontal component of force,