## Vectors involved in the Polarisation Identity

## Clifford Algebra

### The Inner Product

We start with the geometric product of the two vectors **u** and **v**:

Exchanging the vectors u and v gives the relation:

By adding the two equations we get the scalar product:

Or in a more general form:

The relation of geometric products of the ortogonormal basic vectors, can be written in the form of a clifford product as follows:

Or in a more general form with the metric tensor when the base vectors are not orthogonal.

If we square the sum of two vectors (using the geometric product) we get:

(1)

Reordering the elements gives:

(2)

If we square the difference of two vectors we get:

(3)

Reordering the elements gives:

(4)

Summing up equation (2) and (4) gives equation (5):

(5)

We now have the following relations for the scalar product of two vectors:

### The Outer Product

The outer product can be defined in dependence from the geometric product as follows:

Using the following product relation:

(6)

Gives the following relation for the outer product:

(7)

Using the following product relation:

(8)

Gives a further relation for the outer product:

(9)

The diverens between the given two relations for the outer product results in the folowing expression:

(10)

We now have the following relations for the outer product of two vectors:

### Geometric Product

When we sum up the results for the inner (5) and the outer product (10) we come back to the geometric product: