Vectors

A vector is a quantity that has direction as well as size. That is what sets it apart from an ordinary number, or scalar, which has size only.

Directed line segment

Take two different points A, B in the plane. We can join them with a line segment. We usually call that segment AB, but the order of the points does not matter: line segment AB is exactly the same as line segment BA.

That is not always what we want, so we introduce the idea of a directed line segment, in which we distinguish between the start point and the end point. The directed line segment AB starts at A and ends at B. In a diagram we show a directed line segment with an arrow, and in text we write it with an arrow over the points: $\overrightarrow{AB}$. For example:

Line segment AB is shown in blue and directed line segment $\overrightarrow{CD}$ in red. If a directed line segment $\overrightarrow{AB}$ has A = B, we call it the zero directed line segment. Its start point and end point are the same, so you would hardly call it a line segment at all, since it is really just a single point.

The magnitude of a directed line segment is the same as the length of the ordinary line segment: $|\overrightarrow{AB}|=|AB|$.

Direction of a directed line segment

Look at the following two directed line segments $\overrightarrow{AB}$ and $\overrightarrow{CD}$:

At a glance it is clear that the two directed line segments point the same way. We could define “direction” properly, but we will stick with the intuitive idea. Put simply, two directed line segments have the same direction if they are parallel and neither one points the opposite way to the other. Each of these pairs of directed line segments has the same direction:

The last pair, in green, is the directed line segments $\overrightarrow{IJ}$ and $\overrightarrow{IK}$. Here are some pairs of directed line segments that do not have the same direction:

What is a vector

Let’s go back to the picture of two directed line segments with the same direction:

These directed line segments have not only the same direction but also the same magnitude. The only difference is that they sit in different places. Directed line segments that have the same direction and the same magnitude represent one and the same vector.

So a single vector can be drawn in the plane (or in space) in infinitely many ways: we just take any directed line segment with the same direction and magnitude as the vector we want. All of the following directed line segments therefore represent a single vector:

A vector is determined only by its magnitude and direction. The magnitude and direction of a vector are those of any directed line segment that represents it. We usually write vectors in bold with an arrow, just like a directed line segment. So we can have the vector $\overrightarrow{\mathbf{AB}}$ or, more briefly, the vector $\overrightarrow{\mathbf{u}}$.

One particular directed line segment is called a representative of the vector. In the previous picture there are 7 directed line segments in total, and each one is a different representative of the vector. The directed line segments $\overrightarrow{AB}$ and $\overrightarrow{MN}$ represent the same vector, say $\overrightarrow{\mathbf{v}}$, but they are two different representatives of $\overrightarrow{\mathbf{v}}$.

Parallel vectors

Take two vectors $\overrightarrow{\mathbf{u}},\overrightarrow{\mathbf{v}}$. We say that these vectors are parallel (or collinear) if they are parallel as lines. The vectors $\overrightarrow{\mathbf{u}},\overrightarrow{\mathbf{v}}$ need not have the same magnitude. Their direction must be either the same or opposite. Examples of parallel vectors:

All the red vectors are parallel to each other. The direction does not matter any further: it makes no difference whether a vector points to the top right or to the bottom left. The blue vector $\overrightarrow{GH}$ is not parallel to them.

If parallel vectors have the same direction, we say they are in the same direction; if their directions are opposite, they are in opposite directions. Going back to the previous picture, the vectors $\overrightarrow{AB}$ and $\overrightarrow{EF}$ are in the same direction, while the vectors $\overrightarrow{AB}$ and $\overrightarrow{KL}$ are in opposite directions.

Coordinates of vectors

We already know that one vector can be shown by infinitely many directed line segments. One of them starts at the origin of the coordinate system: at the point (0, 0) in the plane, or (0, 0, 0) in space. Look at the following picture:

The directed line segments $\overrightarrow{AB}$ and $\overrightarrow{CD}$ have the same direction and the same magnitude, so they are one vector. The segment $\overrightarrow{CD}$ starts at the origin, so the coordinates of the point C are (0, 0). The same holds for the pair $\overrightarrow{EF}$ and $\overrightarrow{KL}$.

If we know that the segments $\overrightarrow{CD},\overrightarrow{EF}$ start at the origin, we can agree to describe these vectors by their end points only. The start point is the same (the origin), so to tell $\overrightarrow{CD},\overrightarrow{EF}$ apart it is enough to write down their end points.

We say that the vector $\overrightarrow{AB}$, and therefore also the vector $\overrightarrow{CD}$, has coordinates (3, 1), which are the coordinates of the point D, the end point of $\overrightarrow{CD}$. The vector $\overrightarrow{KL}$ then has coordinates (−1, 4), the coordinates of the point F.

How to work out the coordinates of a vector

Now for the mathematical way of finding the coordinates of a vector. We start from the coordinates of the points A, B that define the vector $\overrightarrow{\mathbf{u}}$. So let the vector $\overrightarrow{\mathbf{u}}$ be defined by the points A(2, 3) and B(5, 4). It is the red vector nearer the top in the previous picture. How do we find the coordinates of the point D, the end point of the directed line segment that represents $\overrightarrow{\mathbf{u}}$ and starts at the origin?

We simply move the whole segment $\overrightarrow{AB}$ to the origin. To bring the point A to the origin, we have to move it 2 squares to the left and 3 squares down. We do the same with the point B. In other words, we subtract the coordinates of A from the coordinates of both A and B. That moves the segment $\overrightarrow{AB}$ to where the segment $\overrightarrow{CD}$ is drawn. We can write:

$$\begin{array}{rcl}A-A=(2,3)-(2,3)& =& (0,0)\\ B-A=(5,4)-(2,3)& =& (3,1)\end{array}$$

The coordinates of the vector $\overrightarrow{\mathbf{u}}$ are therefore (3, 1), the end point of the directed line segment that starts at the origin.

In general, if the vector $\overrightarrow{\mathbf{u}}$ is defined by the points A(x₁,y₁) and B(x₂, y₂), we find the coordinates of $\overrightarrow{\mathbf{u}}$ like this:

$$B-A=(x_2-x_1,y_2-y_1)$$

We write the coordinates of the vector $\overrightarrow{\mathbf{u}}$ in round brackets, so the coordinates of the previous vector $\overrightarrow{\mathbf{u}}$ would be written like this: $\overrightarrow{\mathbf{u}}=(3,1)$.

We can also go back to the picture and work out the coordinates of the vector $\overrightarrow{KL}$. The coordinates of the points K and L are K(−2, 1), L(−3, 5). The coordinates of the vector are therefore

$$L-K=(-3,5)-(-2,1)=(-1,4)$$

We can see that this correctly gives the coordinates of the point F.

How to work out the magnitude of a vector

Once we know the coordinates of a vector, we can easily work out its magnitude. Using Pythagoras’ theorem it is easy to show that the magnitude of a vector with coordinates $\overrightarrow{\mathbf{u}}=(u_1,u_2)$ in the plane is

$$|\overrightarrow{\mathbf{u}}|=\sqrt{u_1^2+u_2^2}$$

In the following picture you can see that the coordinates of the vector match the lengths of the sides of the triangle we have drawn, and that by applying Pythagoras’ theorem we really can quickly find the magnitude of the vector.

For the vector $\overrightarrow{\mathbf{u}}=(2,4)$, the magnitude is

$$|\overrightarrow{\mathbf{u}}|=\sqrt{u_1^2+u_2^2}=\sqrt{2^2+4^2}=\sqrt{4+16}=\sqrt{20}$$

Here are two examples for practice.

1. Find the coordinates of the vector u defined by the points A(3, 7), B(−1, 0). The method is completely straightforward: $\overrightarrow{\mathbf{u}}=(u_1,u_2)=B-A$.

   $$u_1=-1-3=-4,u_2=0-7=-7$$

   So the vector has coordinates $\overrightarrow{\mathbf{u}}=(-4,-7)$.

2. The vector $\overrightarrow{\mathbf{u}}$ has coordinates $\overrightarrow{\mathbf{u}}=(2,3)$ and is defined by the points A(6, 6) and B(x₁, y₁), that is, $\overrightarrow{\mathbf{u}}=\overrightarrow{AB}$. Find the coordinates of the point B.

   Once again we set up the familiar equation and make the unknown the subject:

   $$\begin{array}{rcl}{u}_1& =& x_1-6\\ x_1& =& u_1+6\\ x_1& =& 2+6\\ x_1& =& 8\end{array}$$

   We find the second coordinate in exactly the same way:

   $$\begin{array}{rcl}{u}_2& =& y_1-6\\ y_1& =& u_2+6\\ y_1& =& 3+6\\ y_1& =& 9\end{array}$$

   The point B has coordinates (8, 9).