Maths - Multivector Multiplication Types

Here I have tabulated the algebraic rules for the various Grassmann and Clifford vector and scalar multiplications. This allows us to work out higher order products. Therefore any term can be calculated by recursively applying these rules.

  A /\ B A \/ B lc(A,B) rc(A,B) A * B
vector product A /\ B (A ≠ B)
0 (A = B)
bil(A,B) bil(A,B) bil(A,B) (A /\ B)+bil(A,B)
commute B /\ A == - A /\ B B \/ A == bil^t(A,B) lc(A,B) == bil^t(A,B) rc(A,B) == bil^t(A,B) -(A /\ B)+bil^t(A,B)
common factors eij /\ eik
== 0
eij \/ eik
== e1 \/ eijk
lc(eij,eik)
== lc(e1,eijk)
rc(eij,eik)
== rc(e1,eijk)
eij * eik
==
non-common factors eij /\ ekm
== eijkm
eij \/ ekm
== 0
lc(eij,ekm)
== 0
rc(eij,ekm)
== 0
eij * ekm
== eijkm
square

A /\ A == 0
a /\ a == a²

A \/ A == bil(A,A)
a \/ a == 0
lc(A,A) == bil(A,A)
lc(a,a) == 0
rc(A,A) == bil(A,A)
rc(a,a) == 0
A*A==bil(A,A)
a * a == a²
interaction with scalar a /\ A == aA a /\ A == 0 lc(a,A) == aA
lc(A,a) == 0
rc(a,A) == 0
rc(A,a) == aA
a * A == aA
scalar distribution a(A /\ B) == (aA) /\ B
A /\ (aB)
a(A \/ B) == a bil(A,B) a(A \/ B) == a bil(A,B) a(A \/ B) == a bil(A,B) a(A * B) == (aA) * B
A * (aB)
exterior distribution A /\ (B /\ C) == A/\B/\C A /\ (B \/ C) A /\ lc(B,C) A /\ rc(B,C) A /\ (B * C)
unit element

1 (scalar)
1 /\ A = A

pseudoscalar
pseudo \/ A = A

left: 1
right: pseudoscalar

left: pseudoscalar
right: 1
1 (scalar)
1 * A = A
geometric interpretation part with no common factors
union (exclusive-or)
join
part with common factors
intersection
meet
     
associative yes yes no no yes
grade grade(A /\ B) == 2 grade(A \/ B) == grade(lc(A,B)) == grade(rc(A,B)) == grade(A * B) ==
           

where:

Common Factor Theorem

(x /\ y) \/ (u /\ v) == (x /\ y /\ v) \/ u - (x /\ y /\ u) \/ v

Duality

reversion ~

  concept dual
  /\ \/
  grade(A) = m grade(dual(A)) = n - m
  1 pseudoscalar

where:

n = dimension of vector space

Join and Meet

These are two operations associated with geometric intersection and union of spaces, they are denoted by:

  Symbol Spaces
Meet /\ intersectionintersection
Join \/ union U

Confusingly meet and join often use the same symbols '/\' and '\/' as the inner and outer products although the results are slightly different. (some books such as Doran and Lasenby invert this and use \/ for meet and /\ for join so we have to be very careful with terminology and notation).

The mathematical structure of meet and join is an example of a lattice,

geometric Lattice

A line on this diagram means that the element at the bottom of the line is a direct factor of the element at the top. There is not a line from e1 to e123 because we don't need to include e1 when e1 is already a factor of e12 and e31.

To calculate the meet we take a line upwards from both operands until we get to the lowest common denominator.

Meet /\ 1 e1 e2 e3 e12 e31 e23 e123
1 1 e1 e2 e3 e12 e31 e23 e123
e1 e1 e1 e12 e31 e12 e31 e123 e123
e2 e2 e12 e2 e23 e12 e123 e23 e123
e3 e3 e31 e23 e3 e123 e31 e23 e123
e12 e12 e12 e12 e123 e12 e123 e123 e123
e31 e31 e31 e123 e31 e123 e31 e123 e123
e23 e23 e123 e23 e23 e123 e123 e23 e123
e123 e123 e123 e123 e123 e123 e123 e123 e123

To calculate the join we take a line downwards from both operands until we get to the highest common factor.

Join \/ 1 e1 e2 e3 e12 e31 e23 e123
1 1 1 1 1 1 1 1 1
e1 1 e1 1 1 e1 e1 1 e1
e2 1 1 e2 1 e2 1 e2 e2
e3 1 1 1 e3 1 e3 e2 e3
e12 1 e1 e2 1 e12 e1 e2 e12
e31 1 e1 1 e3 e1 e31 e3 e31
e23 1 1 e2 e2 e2 e3 e23 e23
e123 1 e1 e2 e3 e12 e31 e23 e123

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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:

 

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