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Maths - 3D Clifford Algebra - Inner Products

There are different definitions of the inner product as follows:

• Dot product  
•H Hestines inner product. Like the dot product except that it is zero whenever one of its arguments is a scalar.
left contraction left contraction inner product  
right contraction right contraction inner product  

Dot product table:

a•b
 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

Hestines inner product table:

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

left contraction inner product table:

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

right contraction inner product table:

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

 

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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.

cover 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.

cover 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.

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