Higher Engineering Mathematics

226 VECTOR GEOMETRY

Thus,oarepresents a vector quantity, butoais
the magnitude of the vectoroa. Also, positive angles
are measured in an anticlockwise direction from a
horizontal, right facing line and negative angles in a
clockwise direction from this line—as with graph-
ical work. Thus 90◦ is a line vertically upwards
and− 90 ◦is a line vertically downwards.
The resultant of adding two vectors together, say
V 1 at an angleθ 1 andV 2 at angle (−θ 2 ), as shown in
Fig. 21.3(a), can be obtained by drawingoato rep-
resentV 1 and then drawingarto representV 2. The
resultant ofV 1 +V 2 is given byor. This is shown in
Fig. 21.3(b), the vector equation beingoa+ar=or.
This is called the‘nose-to-tail’ methodof vector
addition.

Figure 21.3

Alternatively, by drawing lines parallel toV 1 andV 2
from the noses ofV 2 andV 1 , respectively, and letting
the point of intersection of these parallel lines beR,
givesORas the magnitude and direction of the resul-
tant of addingV 1 andV 2 , as shown in Fig. 21.3(c).
This is called the‘parallelogram’ methodof vector
addition.

Problem 1. A force of 4 N is inclined at an

angle of 45◦to a second force of 7 N, both forces

acting at a point. Find the magnitude of the resul-

tant of these two forces and the direction of the

resultant with respect to the 7 N force by both

the ‘triangle’ and the ‘parallelogram’ methods.

The forces are shown in Fig. 21.4(a). Although the
7 N force is shown as a horizontal line, it could have
been drawn in any direction.

Figure 21.4

Using the ‘nose-to-tail’ method, a line 7 units

long is drawn horizontally to give vector oa in

Fig. 21.4(b). To the nose of this vectoraris drawn

4 units long at an angle of 45◦tooa. The resul-

tant of vector addition isorand by measurement

is10.2 units long and at an angle of 16◦to the

7N force.

Figure 21.4(c) uses the‘parallelogram’ method

in which lines are drawn parallel to the 7 N and 4 N

forces from the noses of the 4 N and 7 N forces,

respectively. These intersect atR. VectorORgives

the magnitude and direction of the resultant of vector

addition and as obtained by the ‘nose-to-tail’ method

is10.2 units long at an angle of 16◦to the 7 N

force.

Problem 2. Use a graphical method to deter-

mine the magnitude and direction of the resultant

of the three velocities shown in Fig. 21.5.

Figure 21.5