Higher Engineering Mathematics

Vector geometry

D

21

Vectors, phasors and the combination

of waveforms

21.1 Introduction

Some physical quantities are entirely defined by
a numerical value and are called scalar quanti-
tiesorscalars. Examples of scalars include time,
mass, temperature, energy and volume. Other phys-
ical quantities are defined by both a numerical value
and a direction in space and these are calledvector
quantitiesorvectors. 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 mag-
nitude 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 byoa=20 m/s at
45 ◦to the horizontal.

Figure 21.1

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, whereA

is the starting point andBthe end point of the

vector,

(iii) a line over the top of letters, e.g.ABora ̄

(iv) letters with an arrow above, e.g. a,A^

(v) underlined letters, e.g.a

(vi)xi+jy, whereiandjare axes at right-angles to

each other; for example, 3i+ 4 jmeans 3 units

in theidirection and 4 units in thejdirection,

as shown in Fig. 21.2.

4

j

0 3 i

A(3,4)

Figure 21.2

(vii) a column matrix

(

a

b

)

; for example, the vector

OAshown in Fig. 21.2 could be represented

by

(

3

4

)

Thus, in Fig. 21.2,

OA≡

−→

OA≡OA≡ 3 i+ 4 j≡

(

3

4

)

The one adopted in this text is to denote vector

quantities inbold print.