254 Higher Engineering Mathematics
Thus, the resultant of the two force vectors is 18 N at
34 ◦ to the 15 N force.
10 N
R
15 N
Figure 24.10
Problem 3. Velocities of 10 m/s, 20 m/s and
15 m/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
10 m/s
20 m/s
15 m/s
Figure 24.11
When more than two vectors are being added the ‘noseto-tail’ method is used.
The order in which the vectors are added does not
matter. In this case the order taken is v 1 , then v 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 0a in Fig. 24.12
(ii) From the nose of v 1 , v 2 is drawn 20 units long at
an angle of 90 ◦ to the horizontal, shown as ab
(iii) From the nose of v 2 , v 3 is drawn 15 units long at
an angle of 195 ◦ to the horizontal, shown as br
(iv) The resultant velocity is given by length 0r and
is measured as 22 m/s and the angle measured to
the horizontal is 105 ◦ .
Thus, the resultant of the three velocities is 22 m/s at
105 ◦ to the horizontal.
b
195Њ
105Њ
3 0 Њ
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 demonstrate 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 vector F is shown in Fig. 24.13 at angle θ 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 24.13
The two components usually taken are a horizontal
component and a vertical component.
If a right-angled triangle is constructed as shown in
Fig. 24.14, then 0a is called the horizontal component
of F and ab is called the vertical component of F.
0
a
F
b
Figure 24.14
Thus, the resultant of the two force vectors is 18 N at
34 ◦ to the 15 N force.
10 N
R
15 N
Figure 24.10
Problem 3. Velocities of 10 m/s, 20 m/s and
15 m/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
10 m/s
20 m/s
15 m/s
Figure 24.11
When more than two vectors are being added the ‘noseto-tail’ method is used.
The order in which the vectors are added does not
matter. In this case the order taken is v 1 , then v 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 0a in Fig. 24.12
(ii) From the nose of v 1 , v 2 is drawn 20 units long at
an angle of 90 ◦ to the horizontal, shown as ab
(iii) From the nose of v 2 , v 3 is drawn 15 units long at
an angle of 195 ◦ to the horizontal, shown as br
(iv) The resultant velocity is given by length 0r and
is measured as 22 m/s and the angle measured to
the horizontal is 105 ◦ .
Thus, the resultant of the three velocities is 22 m/s at
105 ◦ to the horizontal.
b
195Њ
105Њ
3 0 Њ
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 demonstrate 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 vector F is shown in Fig. 24.13 at angle θ 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 24.13
The two components usually taken are a horizontal
component and a vertical component.
If a right-angled triangle is constructed as shown in
Fig. 24.14, then 0a is called the horizontal component
of F and ab is called the vertical component of F.
0
a
F
b
Figure 24.14