252 Higher Engineering Mathematics
An arrow is used to denote the sense, or direction, of the
vector.
The arrow end of a vector is called the ‘nose’ and the
other end the ‘tail’.
For example, a force of 9 N acting at 45 ◦ to the horizontal
is shown in Fig. 24.1.
Note that an angle of + 45
◦ is drawn from the horizontal
and moves anticlockwise.
9 N
0
a
458
Figure 24.1
A velocity of 20 m/s at −60
◦ is shown in Fig. 24.2.
Note that an angle of −60 ◦ is drawn from the horizontal
and moves clockwise.
60Њ
20 m/s
0
b
Figure 24.2
Representing a vector
There are a number of ways of representing vector
quantities. These include:
1. Using bold print
2.
− →
AB where an arrow above two capital letters
denotes the sense of direction, where A is the
starting point and B the end point of the vector
3. AB or a i.e. a line over the top of letters
4. a i.e. an underlined letter
The force of 9N at 45 ◦ shown in Fig. 24.1 may be
represented as:
0a or
− →
0a or 0a
The magnitude of the force is 0a
Similarly, the velocity of 20 m/s at −60 ◦ shown in
Fig. 24.2 may be represented as:
0b or
− →
0b or 0b
The magnitude of the velocity is 0b
In this chapter a vector quantity is denoted by bold
print.
24.4 Addition of vectors by drawing
Adding two or more vectors by drawing assumes that
a ruler, pencil and protractor are available. Results
obtained by drawing are naturally not as accurate as
those obtained by calculation.
(a) Nose-to-tail method
Two force vectors, F 1 and F 2 , are shown in Fig. 24.3.
When an object is subjected to more than one force,
the resultant of the forces is found by the addition of
vectors.
F 2
F 1
Figure 24.3
To add forces F 1 and F 2 :
(i) Force F 1 is drawn to scale horizontally, shown as
0a in Fig. 24.4.
(ii) From the nose of F 1 , force F 2 is drawn at angle
θ to the horizontal, shown as ab.
(iii) The resultant force is given by length 0b, which
may be measured.
This procedure is called the ‘nose-to-tail’ or ‘triangle’
method.
F 2
F 1
a
b
0
Figure 24.4
(b) Parallelogram method
To add the two force vectors, F 1 and F 2 , of Fig. 24.3:
(i) A line cb is constructed which is parallel to and
equal in length to 0a (see Fig. 24.5).
An arrow is used to denote the sense, or direction, of the
vector.
The arrow end of a vector is called the ‘nose’ and the
other end the ‘tail’.
For example, a force of 9 N acting at 45 ◦ to the horizontal
is shown in Fig. 24.1.
Note that an angle of + 45
◦ is drawn from the horizontal
and moves anticlockwise.
9 N
0
a
458
Figure 24.1
A velocity of 20 m/s at −60
◦ is shown in Fig. 24.2.
Note that an angle of −60 ◦ is drawn from the horizontal
and moves clockwise.
60Њ
20 m/s
0
b
Figure 24.2
Representing a vector
There are a number of ways of representing vector
quantities. These include:
1. Using bold print
2.
− →
AB where an arrow above two capital letters
denotes the sense of direction, where A is the
starting point and B the end point of the vector
3. AB or a i.e. a line over the top of letters
4. a i.e. an underlined letter
The force of 9N at 45 ◦ shown in Fig. 24.1 may be
represented as:
0a or
− →
0a or 0a
The magnitude of the force is 0a
Similarly, the velocity of 20 m/s at −60 ◦ shown in
Fig. 24.2 may be represented as:
0b or
− →
0b or 0b
The magnitude of the velocity is 0b
In this chapter a vector quantity is denoted by bold
print.
24.4 Addition of vectors by drawing
Adding two or more vectors by drawing assumes that
a ruler, pencil and protractor are available. Results
obtained by drawing are naturally not as accurate as
those obtained by calculation.
(a) Nose-to-tail method
Two force vectors, F 1 and F 2 , are shown in Fig. 24.3.
When an object is subjected to more than one force,
the resultant of the forces is found by the addition of
vectors.
F 2
F 1
Figure 24.3
To add forces F 1 and F 2 :
(i) Force F 1 is drawn to scale horizontally, shown as
0a in Fig. 24.4.
(ii) From the nose of F 1 , force F 2 is drawn at angle
θ to the horizontal, shown as ab.
(iii) The resultant force is given by length 0b, which
may be measured.
This procedure is called the ‘nose-to-tail’ or ‘triangle’
method.
F 2
F 1
a
b
0
Figure 24.4
(b) Parallelogram method
To add the two force vectors, F 1 and F 2 , of Fig. 24.3:
(i) A line cb is constructed which is parallel to and
equal in length to 0a (see Fig. 24.5).