vectors

continuation of vectors

Vectors, phosphors and the combination
of waveform
21.1 Introduction
Some physical quantities are entirely defined by
a numerical value and are called scalar quantities
or scalars. Examples of scalars include time,
mass, temperature, energy and volume. Other physical
quantities are defined by both a numerical value
and a direction in space and these are called vector
quantities or vectors. Examples of vectors include
force, velocity, moment and displacement.
21.2 Vector addition
A vector may be represented by a straight line, the
length of line being directly proportional to the magnitude
of the quantity and the direction of the line
being in the same direction as the line of action of
the quantity. An arrow is used to denote the sense
of the vector, that is, for a horizontal vector, say,
whether it acts from left to right or vice-versa. The
arrow is positioned at the end of the vector and this
position is called the ‘nose’of the vector. Figure 21.1
shows a velocity of 20 m/s at an angle of 45◦ to the
horizontal and may be depicted by oa=20 m/s at
45◦ to the horizontal.
To distinguish between vector and scalar quantities,
various ways are used. These include:
(i) bold print,
(ii) two capital letters with an arrow above them to
denote the sense of direction, e.g.
−→
AB, where A
is the starting point and B the end point of the
vector,
(iii) a line over the top of letters, e.g. AB or ¯a
(iv) letters with an arrow above, e.g. 
a, 
A
(v) underlined letters, e.g. a
(vi) xi+jy, where i and j are axes at right-angles to
each other; for example, 3i+4j means 3 units
in the i direction and 4 units in the j direction.
4
j
0 3 i
A(3,4)
(vii) a column matrix ab
; for example, the vector
OA shown could be represented
OA ≡
−→
OA ≡ OA ≡ 3i + 4j ≡ 34

The one adopted in this text is to denote vector
quantities in bold print.
226 VECTOR GEOMETRY
Thus, oa represents a vector quantity, but oa is
the magnitude of the vector oa. 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 graphical
work. Thus 90◦ is a line vertically upwards
and −90◦ is a line vertically downwards.
The resultant of adding two vectors together, say
V1 at an angle θ1 and V2 at angle (−θ2), as shown in
Fig. 21.3(a), can be obtained by drawing oa to represent
V1 and then drawing ar to represent V2. The
resultant of V1 +V2 is given by or. This is shown in
Fig. 21.3(b), the vector equation being oa+ar=or.
This is called the ‘nose-to-tail’ method of vector
addition.
Alternatively, by drawing lines parallel to V1 and V2
from the noses of V2 and V1, respectively, and letting
the point of intersection of these parallel lines be R,
givesORas the magnitude and direction of the resultant
of adding V1 and V2, as shown in Fig. 21.3(c).
This is called the ‘parallelogram’method of vector
addition.
Problem 1. A force of 4N 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 resultant
of these two forces and the direction of the
resultant with respect to the 7N force by both
the ‘triangle’ and the ‘parallelogram’ methods.
The forces are shown in Fig. 21.4(a). Although the
7N 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
 To the nose of this vector ar is drawn
4 units long at an angle of 45◦ to oa. The resultant
of vector addition is or and by measurement
is 10.2 units long and at an angle of 16◦ to the
7N force.
we  use the ‘parallelogram’ method
in which lines are drawn parallel to the 7N and 4N
forces from the noses of the 4N and 7N forces,
respectively. These intersect at R. Vector OR gives
the magnitude and direction of the resultant of vector
addition and as obtained by the ‘nose-to-tail’method
is 10.2 units long at an angle of 16◦ to the 7N
force.