# Maths - Comparison of 2D multivector with Quaternion

So how does this algebra compare with other algebras? Since it has 4 'dimensions' lets compare it with quaternions. Both these algebras implement addition by adding corresponding terms, as usual with this type of algebra, the differences (if any) are in the multiplication rules:

The 2D multivector multiplication table is:

 a*b b.e b.e1 b.e2 b.e12 a.e 1 e1 e2 e12 a.e1 e1 1 e12 e2 a.e2 e2 -e12 1 -e1 a.e12 e12 -e2 e1 -1

The multiplication table for quaternions is:

 a*b b.1 b.i b.j b.k a.1 1 i j k a.i i -1 k -j a.j j -k -1 i a.k k j -i -1

These tables are very similar, there is a difference in the 'square' terms, i.e. the terms on the leading diagonal. In the case of 2D multivector there are 3 positives and 1 negative, in the case of the quaternion there is 1 positive and 3 negatives. Otherwise the tables seem very similar, in both cases the real terms commute and the other terms (not on the leading diagonal) anticommute.

Does anyone know if:

• Is there a way to relate these two types?
• Quaternions always have an inverse, do 2D multivectors always have an inverse?

## Comparison of 2D multivector with Complex Numbers

We can relate 2D multivectors with complex numbers.

If we let c be a complex number:

c = a + i b

where a and b are the real and imaginary parts.

Assume that we can also represent a complex number by a linear sum of two basis vectors:

x = a e1 + b e2

When we chek this out, using the multipication rules above, this does not quite work. for instance squaring e2 gives +1 not -1.

However if we multiply both sides by e1 we get:

e1 x = a + b e1e2

This has the correct properties, for instance, e1e2 is equivilant to:

e1e2 = √-1

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

Other Math Books