## Arbitrary Vectors

So far, we have only considered the possibility that we can kick the ball either to the left or to the right. In general, of course, we can kick the ball in many more directions, i.e. we can kick it in the air.

###### Figure 1.3.1

We are now considering that the ball can be kicked in two dimensions rather than simply one. (Of course, in real life, the ball is in fact kicked in three dimensions, but thankfully such three dimensional motion is not on the syllabus!)

However, how do we now state what the direction of the vector is? Before we could simply say that it was pointing left or right (or west or east), but what about if it is pointing in an arbitrary direction?

###### Figure 1.3.2

It seems only natural to define the direction of a vector using angles!

But an angle relative to what? An angle requires two lines in order to exist at all.

###### Figure 1.3.3

Of course, if we kick a ball from the ground, it makes perfect sense to define the direction as the angle between the vector and the ground itself.

###### Figure 1.3.4

But what if we are throwing the ball downwards?

###### Figure 1.3.5

In general, to define the direction of an arbitrary vector, we introduce what are known as *axes *(plural of axis). The axes you are likely most familiar with, and the only one we shall use in this subject, are the \(x\)-\(y\) axes.

###### Figure 1.3.6

This is composed of two axes (an \(x\) axis and a \(y\) axis) which are perpendicular to each other.

### Exam Tip

If the angle between two lines is \(90^{\circ}\), then those lines are said to be *perpendicular*, *orthogonal*, *normal* or at *right angles* to each other.

Any of these terms may be used in your exam.

Note that, according to this definition, the following is also an \(x\)-\(y\) axes. It is also composed of two axes (an \(x\) axis and a \(y\) axis) which are perpendicular to each other.

###### Figure 1.3.7

In general, we are in fact free to rotate an \(x\)-\(y\) axes before we use it in a particular question. As long as the axes are at right angles to each other, they are considered an \(x\)-\(y\) axes.

#### \(x\)-\(y\) axes

#### \(x\)-\(y\) axes

#### Not \(x\)-\(y\) axes

###### Figure 1.3.8

This will be useful in future topics when we look at the motion of objects on inclined planes, e.g. hills. For now, we only need to use the \(x\)-\(y\) axes that we are most familiar with, namely with the \(x\) axis pointing in the east-west direction direction and the \(y\) axis in the north-south direction.

###### Figure 1.3.9

Now, if we kick a ball to the right, we can be more “mathsy” and instead say the ball was kicked in the “positive \(x\) direction”.

###### Figure 1.3.10

If we instead kicked it in the opposite direction, we would say it was kicked in the “negative \(x\) direction”.

###### Figure 1.3.11

Similarly, if we throw a ball vertically upwards, it was thrown in the “positive \(y\) direction”.

###### Figure 1.3.12

And if we throw it vertically downwards, it was thrown in the “negative \(y\) direction”.

###### Figure 1.3.13

Now, how does this help us with more arbitrary directions? Well, we now have some lines we can use to make our angle!

You may be wondering though, don’t we have too many lines? And you’d be right! We only need two lines and yet we seem to have three – the \(x\) axis, the \(y\) axis and the vector itself.

###### Figure 1.3.14

So, which of the two do we choose? Obviously, as we are trying to define the direction of the vector, we need to use the vector itself as one of the two lines. If we just use the \(x\) and \(y\) axes, the angle will always be \(90^{\circ}\) and we will have learned nothing about the direction of our vector.

###### Figure 1.3.15

So, we should instead choose the vector and one of the axes. But which one? The answer is: it depends on the question, but neither is wrong!

For example, if a question starts, “a ball is kicked at an angle of \(30^{\circ}\) relative to the horizontal ground”, then it makes sense to use the \(x\) axis as our second line, since we can draw the \(x\) axis along the ground.

###### Figure 1.3.16

Then, we would say the direction the ball is kicked was “\(30^{\circ}\) relative to the positive \(x\) axis”.

### Cause of Confusion

Of course, the statement “the ball is kicked at \(30^{\circ}\) relative to the positive \(x\)-axis” could refer to two different angles.

###### Figure 1.3.17

By convention, unless it is explicitly stated, the vector is assumed to be the one created in the *anti-clockwise* direction (*from* the axis *towards* the vector).

However, we can equally say that the ball was kicked “\(120^{\circ}\) relative to the negative \(y\) axis”

###### Figure 1.3.18

or “\(210^{\circ}\) relative to the negative \(x\) axis”

###### Figure 1.3.19

or even “\(300^{\circ}\) relative to the positive \(y\) axis”

###### Figure 1.3.20

They all mean the exact same thing! However, intuitively, it makes the most sense to choose the first option in this example as it is the easiest to visualise.

There will also, however, be questions that instead specify the direction relative to the \(y\) axis. For example, a question may instead start, “a ball is thrown at an angle of \(30^{\circ}\) relative to a wall as shown”. Then, it makes the most intuitive sense to define the angle as “\(30^{\circ}\) relative to the negative \(y\) axis”.

###### Figure 1.3.21

Now that we know how to state a vector’s direction using an angle, we can now fully describe a vector without the need to draw a diagram!

For example, the statement “a vector has a magnitude of \(5\) units and is at angle of \(30^{\circ}\) relative to the positive \(x\) axis” completely describes a vector. There is no additional information we could obtain about this vector if we were instead given it on a diagram.

In fact, we can write the statement “a vector has a magnitude of \(5\) units and is at angle of \(30^{\circ}\) relative to the positive \(x\) axis” more compactly by writing this vector in *polar form*.

In this example, this vector in polar form is given by \(\langle 5 \,\, \angle \,\, 30^{\circ}\rangle\).

### Key Point

A vector \(\vec{a}\) is written in polar form as \(\langle |\vec{a}| \,\, \angle \,\, \theta \rangle\), where \(|\vec{a}|\) is the magnitude of the vector and \(\theta\) is the angle anticlockwise from the positive \(x\) axis to that vector.

Note that the angle that is given in polar form is *always* defined anticlockwise relative to positive \(x\) axis.

For example, let us assume that the magnitude of the below vector is 7 units.

###### Figure 1.3.22

This vector is in fact written in polar form as \(\langle 7 \,\, \angle \,\, 60^{\circ}\rangle\) as the anticlockwise angle between it and the positive \(x\) axis is \(60^{\circ}\).

###### Figure 1.3.23

## Guided Example 1.3.1

Using a ruler and a protractor, draw the following vectors:

(a) a vector of magnitude \(3\) cm at an angle of \(50^{\circ}\) relative to the positive \(x\) axis

(b) a vector of magnitude \(5\) cm at an angle of \(210^{\circ}\) relative to the negative \(y\) axis

(c) \(\langle 4 \mbox { cm} \,\, \angle \,\, 10^{\circ}\rangle\)

(d) \(\langle 6.5 \mbox { cm} \,\, \angle \,\, 130^{\circ}\rangle\)

**(a)** Using a ruler, we represent the vector by drawing an arrow of length \(3\) cm at an angle of \(50^{\circ}\) anti-clockwise from the positive \(x\) axis, as below.

\[\,\]

\[\,\]

**(b)** Likewise, we draw an arrow of length \(5\) cm at an angle of \(210^{\circ}\) anti-clockwise from the negative \(y\) axis, as below.

\[\,\]

\[\,\]

**(c)** This vector is represented by an arrow of length \(4\) cm at an angle of \(10^{\circ}\) anticlockwise relative to to the positive \(x\) axis, as below.

\[\,\]

\[\,\]

**(d)** This vector is represented by an arrow of length \(6.5\) cm at an angle of \(130^{\circ}\) anticlockwise relative to to the positive \(x\) axis, as below.

\[\,\]

### Exam Tip

In general, unless it is specifically stated, you do not need to be precise when drawing the lengths of vectors like in the above example.

Of course, it is often not possible to be precise as vectors can even be kilometres long!

Likewise, vectors do not always represent distances but can represent other quantities like velocity, force etc. as we shall see later.

Instead, what’s more important is that the *relative* sizes of vectors are similar.

For example, if vector \(\vec{a}\) has a magnitude of \(200\) Newtons, draw that vector at any length you wish (in the direction specified). If we then need to draw vector \(\vec{b}\), of magnitude \(400\) Newtons, on that same diagram, you should *roughly* draw that vector twice as long as \(\vec{a}\) (in the direction specified).

Doing this does not gives you marks – instead, it is beneficial to you so you can easily recognise which vector is which if you need to use this diagram again in answering other parts of the question.