Geometric Products
Start with a vector space
Inner:
similar to dot product (is the dot product on vectors)
Symmetric, linear on vectors
Outer:
similar to cross-product
Problem with cross-product:
3D only
Not associative
Outer product: directed area element
Anti-symmetric, linear, assocative on vectors
Geometric product:
For vectors:
ab = a^b + inner(a,b)
Associative, linear, but neither symmetric nor anti-symmetric
ba = b^a+inner(b,a) = -a^b+inner(a,b)
Advantage:
inverse exists
b*inverse(b) = 1
inverse(b) = b / inner(b,b)