In this previous blog, I mentioned that there are two ways in which to multiply vectors, either the dot (scalar) product of the cross (vector) product. The dot product is pretty easy to do. If we have two vectors , the *dot product* is just given by

where is the angle between them.

The *vector (cross) product* is a little more complicated.

Let us suppose our vectors are and that can bet written as and , to find the vector product it is easiest to use matrices. This is the way I have always done it, and the way I teach it to my students, but if anyone reading this has a different method they wish to share that would be great.

So, we write again as and as where the brackets indicate that these are now matrices.

## Using the determinant matrix to calculate the vector product

where the expression on the right is the *determinant matrix*.

To calculate each component we work out the determinant of three separate matrices, as shown below. For the component, we cross out the top line and first column of the matrix, and compute the determinant of the remaining matrix.

For our example, this will be . Similarly, for the component

which will be , but not that we take the

*negative*of this. Finally for the component we have

which will be .

Summarising, we can write . This is the vector product of the two vectors.

In the simple case where and then will simply be

which comes to be .

If we want then we must write

which comes to be . This is shown in the figure below.

on 19/08/2014 at 07:01 |The direction of the angular velocity vector | thecuriousastronomer[…] Writing this in terms of vectors, and remembering that division of vectors is not defined, we instead write that where is the vector product (or cross-product), as I discussed in this blog here. […]