2 The mathematical description of vector quantities

Because a vector has a direction as well as a magnitude we can represent a vector by drawing a line. The length of the line represents the magnitude of the vector given some appropriate scale, and the direction of the line represents the direction of the vector. We call this representation a directed line segment . For example, Figure 3 shows a vector which represents a velocity of $3\text{m}{\text{s}}^{-1}$ north-west. Note that the arrow on the vector indicates the direction required.

More generally, Figure 4 shows an arbitrary vector quantity.

It is important when writing vectors to distinguish them from scalars. Various notations are used. In Figure 4 we emphasise that we are dealing with the vector from $A$ to $B$ by using an arrow and writing $\overrightarrow{AB}$ . Often, in textbooks, vectors are indicated by using a bold typeface such as $a$ . It is difficult when handwriting to reproduce the bold face and so it is conventional to underline vector quantities and write $\underset{̲}{a}$ instead. So $\overrightarrow{AB}$ and $\underset{̲}{a}$ represent the same vector in Figure 4. We can also use the notation $\underset{̲}{AB}$ . In general in this Workbook we will use underlining but we will also use the arrow notation where it is particularly helpful.

Example 1

Figure 5 shows an object pulled by a force of 10 N at an angle of $60^{\circ }$ to the horizontal.

Show how this force can be represented by a vector.

Solution

The force can be represented by drawing a line of length 10 units at an angle of $60^{\circ }$ to the horizontal, as shown below.

We have labelled the force $\underset{̲}{F}$ . When several forces are involved they can be labelled ${\underset{̲}{F}}_1$ , ${\underset{̲}{F}}_2$ and so on.

When we wish to refer simply to the magnitude (or length) of a vector we write this using the modulus sign as $\left|\overrightarrow{AB}\right|$ , or $\left|\underset{̲}{a}\right|$ , or simply $a$ (without the underline.)

In general two vectors are said to be equal vectors if they have the same magnitude and same direction. So, in Figure 7 the vectors $\overrightarrow{CD}$ and $\overrightarrow{AB}$ are equal even though their locations differ.

This is a useful and important property of vectors: a vector is defined only by its direction and magnitude, not by its location in space. These vectors are often called free vectors.

The vector $-\underset{̲}{a}$ is a vector in the opposite direction to $\underset{̲}{a}$ , but has the same magnitude as $\underset{̲}{a}$ , as shown in Figure 8.

Geometrically, if $\underset{̲}{a}=\overrightarrow{AB}$ then $-\underset{̲}{a}=\overrightarrow{BA}$ .

Exercises

1. An object is subject to two forces, one of 3 N vertically downwards, and one of 8 N, horizontally to the right. Draw a diagram representing these two forces as vectors.
2. Draw a diagram showing an arbitrary vector $\underset{̲}{F}$ . On the diagram show the vector $-\underset{̲}{F}$ .
3. Vectors $\underset{̲}{p}$ and $\underset{̲}{q}$ are equal vectors. Draw a diagram to represent $\underset{̲}{p}$ and $\underset{̲}{q}$ .
4. If $\underset{̲}{F}$ is a vector, what is meant by $F$ ?

Answer