Vectors 253
(ii) A line ab is constructed which is parallel to and
equal in length to 0c.
(iii) The resultant force is given by the diagonal of the
parallelogram, i.e. length 0b.
This procedure is called the ‘parallelogram’ method.
0
c
F 1
F 2
a
b
Figure 24.5
Problem 1. A force of 5 N is inclined at an angle
of 45 ◦ to a second force of 8 N, 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 8 N force by:
(a) the ‘nose-to-tail’ method, and (b) the
‘parallelogram’ method.
The two forces are shown in Fig. 24.6. (Although the
8 N force is shown horizontal, it could have been drawn
in any direction.)
458
5 N
8 N
Figure 24.6
(a) ‘Nose-to tail’ method
(i) The 8 N force is drawn horizontally 8 units long,
shown as 0a in Fig. 24.7
(ii) From the nose of the 8 N force, the 5 N force
is drawn 5 units long at an angle of 45 ◦ to the
horizontal, shown as ab
(iii) The resultant force is given by length 0b and
is measured as 12 N and angle θ is measured
as 17 ◦ .
458
5 N
8 N
a
0
b
Figure 24.7
(b) ‘Parallelogram’ method
(i) In Fig. 24.8, a line is constructed which is parallel
to and equal in length to the 8 N force
(ii) A line is constructed which is parallel to and equal
in length to the 5 N force
(iii) The resultant force is given by the diagonal
of the parallelogram, i.e. length 0b, and is
measured as 12 N and angle θ is measured
as 17
◦ .
458
5 N
8 N
b
0
Figure 24.8
Thus, the resultant of the two force vectors in Fig. 24.6
is 12 N at 17 ◦ to the 8 N force.
Problem 2. Forces of 15 N and 10 N 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 15 N force.
15 N
10 N
Figure 24.9
Using the ‘nose-to-tail’ method:
(i) The 15 N force is drawn horizontally 15 units
long as shown in Fig. 24.10
(ii) From the nose of the 15 N force, the 10 N force
is drawn 10 units long at an angle of 90 ◦ to the
horizontal as shown
(iii) The resultant force is shown as R and is measured
as 18 N and angle θ is measured as 34 ◦ .
(ii) A line ab is constructed which is parallel to and
equal in length to 0c.
(iii) The resultant force is given by the diagonal of the
parallelogram, i.e. length 0b.
This procedure is called the ‘parallelogram’ method.
0
c
F 1
F 2
a
b
Figure 24.5
Problem 1. A force of 5 N is inclined at an angle
of 45 ◦ to a second force of 8 N, 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 8 N force by:
(a) the ‘nose-to-tail’ method, and (b) the
‘parallelogram’ method.
The two forces are shown in Fig. 24.6. (Although the
8 N force is shown horizontal, it could have been drawn
in any direction.)
458
5 N
8 N
Figure 24.6
(a) ‘Nose-to tail’ method
(i) The 8 N force is drawn horizontally 8 units long,
shown as 0a in Fig. 24.7
(ii) From the nose of the 8 N force, the 5 N force
is drawn 5 units long at an angle of 45 ◦ to the
horizontal, shown as ab
(iii) The resultant force is given by length 0b and
is measured as 12 N and angle θ is measured
as 17 ◦ .
458
5 N
8 N
a
0
b
Figure 24.7
(b) ‘Parallelogram’ method
(i) In Fig. 24.8, a line is constructed which is parallel
to and equal in length to the 8 N force
(ii) A line is constructed which is parallel to and equal
in length to the 5 N force
(iii) The resultant force is given by the diagonal
of the parallelogram, i.e. length 0b, and is
measured as 12 N and angle θ is measured
as 17
◦ .
458
5 N
8 N
b
0
Figure 24.8
Thus, the resultant of the two force vectors in Fig. 24.6
is 12 N at 17 ◦ to the 8 N force.
Problem 2. Forces of 15 N and 10 N 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 15 N force.
15 N
10 N
Figure 24.9
Using the ‘nose-to-tail’ method:
(i) The 15 N force is drawn horizontally 15 units
long as shown in Fig. 24.10
(ii) From the nose of the 15 N force, the 10 N force
is drawn 10 units long at an angle of 90 ◦ to the
horizontal as shown
(iii) The resultant force is shown as R and is measured
as 18 N and angle θ is measured as 34 ◦ .