Vectors

Vectors are quantities which have both magnitude (ie. size) and direction. The most common examples of these are velocity ($3ms^{-1}$ to the right) and force (10 Newtons pointing vertically down). The easiest way to describe such a quantity is an arrow, where the magnitude gives the length and the direction is given by, well, the direction of the arrow. The important point about this is that the position of the vector itself doesn’t matter. In the figure below we place the same arrow in several different places and they are all the same vector.

A vector, with magnitude given by its length and direction given by the direction of the arrow, placed at different points in the plane. Note that the position of the start of the arrow is now important, just the relationship between the start and the end of the arrow.

We could define a vector by the length and the angle that it makes with the horizontal axis, but in general we define it by how much it goes in the horizontal direction and how much it goes in the vertical direction, that is, how much it goes in the -direction and how much it goes in the -direction. You can see that in figure \ref{vec1} the vectors all go along 2 units in the -direction and up 3 units in the -direction. We write this vector (which we will call ) as . The numbers 2 and 3 are said to be the components of the vector . All of the arrows in the figure correspond to precisely this vector. The starting position of the vector is unimportant. All that is important is how much it goes along and how much it goes up. This can easily be extended to higher dimensions. We can think of the amount that an arrow goes along in the , and directions and this would be labelled by three numbers, for instance where the -2 just means that it goes backwards two units in the direction.

The magnitude of a vector is just the size, without the direction, of the vector, so the magnitude of a vector is a scalar quantity. It is just the same as the Pythagorean length of the vector. So in the case of the vector in figure \ref{vec1} the magnitude is . We denote the magnitude of a vector as and so . The magnitude is always equal to the square root of the sum of the squared components of the vector. I could write a vector, , in 7 dimensions, for instance (not that I could picture it) as and the magnitude would be: .

Now that we can write down a vector using the angle brackets we can ask what happens when we add together two vectors. We can think of adding together their descriptions. For instance, if we have a vector which is and one which was , then corresponds to “go 3 to the right and 4 up, then 5 to the right, then 7 up”, which is the same as 8 to the right, then 11 up which we can write as: . So, adding together vectors is done by simply adding them component by component. If and are vectors in an -dimensional space where and , are just numbers, ie. the components of the vectors then:

It is clear then that if this is how we add vectors together, that it doesn’t matter which one comes first, and so . We can also view this pictorially in the figure here:

The addition of two vectors. We don’t need to put axes here because the position of the vectors are unimportant. When we add them together we can first move along and then or vice versa. The black line is the addition of the two which gives

It turns out that if we know how to add together two vectors then we can work out how to multiply a vector by a scalar. The answer is very simple but we can think about it first just in terms of adding a vector to itself, ie. multiplying it my the scalar 2. If then . In fact:

where is any scalar. This includes negative numbers and so the negative of a vector is just a vector of the same length pointing in the opposite direction. Thus if you are asked to find the difference between two vectors, this is the same as adding one to the negative of the other. This is shown here:

The addition and subtraction of vectors. Subtraction should be thought of as the addition of the negative of the vector. Remember: The position of the vector is unimportant. For addition you simply lay them end to end in whatever order you want.

Keep this in mind at all times: A vector doesn’t have a particular starting point, it is an arrow and can be moved around to any position. It is only the length and direction of the vector which is important.

So, we’ve shown how to find the length of a vector, and so it’s very easy to create a unit vector in a given direction. Let’s say that we have a vector and we want to find a unit vector in this direction. We can do this by simply dividing the vector by its magnitude, which in this case is . So a unit vector in the same direction as is:

Check for yourself that this new vector has magnitude 1. There is another unit vector with the same direction which is the vector pointing in the opposite direction – i.e. . In general a unit vector in the direction of a vector is:

You can show in general that this will have magnitude 1.

So far we have always used the angled bracket notation to write down out vectors but we can write them in another way. To do this we have to define what are called the standard basis vectors. They are normally given by the vectors , and where:

Of course, if we are dealing with higher dimensions then we need to be a bit more creative with the names, but for now, this will do just fine. We can now write any vector as a sum of these 3 standard basis vectors. For instance:

and in general if we have a vector then we can write this as: . As an exercise, we can also see that a unit vector in the direction of is:

Of course this can be generalised to any number of dimensions.