Maths - Clifford Algebra - Contraction

The inner product • is an extension to the idea of the dot product in conventional vector analysis, but the dot product normally combines two vectors to give a scalar quantity, a scalar could be taken as the point at the origin so the 'meet' of two vectors is the point at the origin.

Here the • operation has been extended to higher order elements so we take two bivectors and get a vector.

There are different ways that the dot product can be extended to higher order elements such as bivectors, so we have to be careful to choose the right type to give the correct 'meet' operation.

Contraction Inner Product

A  B is a blade representing the complement (within the subspace B) of the orthogonal projection of A onto B.

Here we will attempt to build up the full multiplication table by taking each blade type in turn. We already know that the dot product of two vectors is a scalar:

b.e1 b.e2 b.e3
a.e1 e 0 0
a.e2 0 e 0
a.e3 0 0 e

αβ = α β

Vector and scalar
a β = 0

Scalar and vector
α b = α b

What is the general case for multipying two bivectors?

b.e12 b.e31 b.e23
a.e12 -e    
a.e31   -e  
a.e23     -e

vector times bivector ?

a  (b ∧ B) = (a  b) ∧ B – b ∧(a  B)

b.e12 b.e31 b.e23
a.e1 e2 -e3 0
a.e2 -e1 0 e3
a.e3 0 e1 -e2

full table ?

b.e b.e1 b.e2 b.e3 b.e12 b.e31 b.e23 b.e123
a.e e e1 e2 e3 e12 e31 e23 e123
a.e1 e1 e 0 0 e2 -e3 0 e23
a.e2 e2 0 e 0 -e1 0 e3 e31
a.e3 e3 0 0 e 0 e1 -e2 e12
a.e12 e12 -e2 e1 0 -e 0 0 -e3
a.e31 e31 e3 0 -e1 0 -e 0 -e2
a.e23 e23 0 -e3 e2 0 0 -e -e1
a.e123 e123 e23 e31 e12 -e3 -e2 -e1 -e


metadata block
see also:


Correspondence about this page

Book Shop - Further reading.

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.

flag flag flag flag flag flag 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.


flag flag flag flag flag flag 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.


Terminology and Notation

Specific to this page here:


This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.