Understanding Engineering Mathematics

and ‘vector’ are also used in matrix theory, and indeed matrices can be used to represent
vectors – a 3×1 column matrix can be used to represent a vector for example. Again, the
connections between these ideas are not always clear. However, we will usually use the
representation of vectors by arrays, which is relatively straightforward to operate.
In this book we will usually denote vectors by bold lower case – in written work it
is usual to denote them by under line (a)oroverline(a). To denote specific vector
displacements from points A to B we use the notation

−→

AB or AB

Note that vectors have units of course, depending on the physical quantity they represent.
Whichever representation of vectors we use, we will eventually need a three dimen-
sional coordinate system – simply so that we can state such things as length and direction
unequivocally. There are a number of such coordinate systems in common use but we
will use the simplest and most popular one – the generalisation ofrectangular Cartesian
coordinates.

Exercise on 11.1

Classify the following as scalar or vector quantities:

(i) area (ii) gravitational force (iii) electric field at a point

(iv) density (v) potential energy (vi) work

(vii) magnetic field (viii) time (ix) pressure

(x) acceleration (xi) voltage (xii) momentum

Answers

(i), (iv), (v), (vi), (viii), (xi) are all scalars. The rest are all vectors.

11.2 Vectors as arrows

This is the most common approach to vectors at the elementary level. We treat the line
segment representing a vectoraas an abstract geometrical ‘free’ object. Ifais specifically
fixed to an origin, then it is referred to as aposition vector– its tip designating the
position of the point at the tip. Pictorially, we might display vectors as in Figure 11.1.

a

−a

Figure 11.1Free vectors as arrows.

Note that the vector−ais simplyain the opposite sense.
An algebra of such objects can be constructed which reflects geometrically the way in
which vector quantities combine. Naturally, two vectors are equal if and only if they have
the same magnitude and direction. Thezero vectorhas zero magnitude and undefined
direction – it is denoted 0.