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’.
Forexample,aforceof9Nactingat45◦tothehorizontal
is shown in Fig. 24.1.
Note that an angle of+ 45 ◦is drawn from the horizontal
and movesanticlockwise.
9N
0
a
458
Figure 24.1
A velocity of 20m/s at− 60 ◦is shown in Fig. 24.2.
Note that an angle of− 60 ◦is drawn from the horizontal
and movesclockwise.
60
20 m/s
0
b
Figure 24.2
Representing a vector
There are a number of ways of representing vector
quantities. These include:
- Usingbold print
2.
−→
AB where an arrow above two capital letters
denotes the sense of direction, where A is the
starting point andBthe end point of the vector
- ABorai.e. a line over the top of letters
- ai.e. an underlined letter
Theforceof9Nat45◦shown in Fig. 24.1 may be
represented as:
0 a or
−→
0 a or 0 a
The magnitude of the force is 0a
Similarly, the velocity of 20m/s at− 60 ◦shown in
Fig. 24.2 may be represented as:
0 b or
−→
0 b or 0 b
The magnitude of the velocity is 0b
In this chapter a vector quantity is denoted bybold
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 andF 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 forcesF 1 andF 2 :
(i) ForceF 1 is drawn to scale horizontally, shown as
0 ain Fig. 24.4.
(ii) From the nose ofF 1 ,forceF 2 is drawn at angle
θto the horizontal, shown asab.
(iii) The resultant force is given by length 0 b,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 andF 2 , of Fig. 24.3:
(i) A linecbis constructed which is parallel to and
equal in length to 0 a(see Fig. 24.5).