Cayley Table for Geometric Algebra
To multiply two multivectors (a * b) then we multiply each part of a by each part of b, so that we have multiplied every combination of terms. When multiplying terms the result will be of type given by the following table:
The entries in the table only shows the type and sign change of the product, it does not show its absolute value. We therefore need to prefix the product by its numerical value which is the real number which is the product of the numbers at the top and left headings.
a*b |
b.e | b.e1 | b.e2 | b.e3 | b.e12 | b.e31 | b.e23 | b.e123 |
| a.e | 1 | e1 | e2 | e3 | e12 | e31 | e23 | e123 |
| a.e1 | e1 | 1 | e12 | -e31 | e2 | -e3 | e123 | e23 |
| a.e2 | e2 | -e12 | 1 | e23 | -e1 | e123 | e3 | e31 |
| a.e3 | e3 | e31 | -e23 | 1 | e123 | e1 | -e2 | e12 |
| a.e12 | e12 | -e2 | e1 | e123 | -1 | e23 | -e31 | -e3 |
| a.e31 | e31 | e3 | e123 | -e1 | -e23 | -1 | e12 | -e2 |
| a.e23 | e23 | e123 | -e3 | e2 | e31 | -e12 | -1 | -e1 |
| a.e123 | e123 | e23 | e31 | e12 | -e3 | -e2 | -e1 | -1 |
This table fully defines how multiplication works, we don't have to bother with all the stuff about ^ that we previously discussed to derive the table if we don't want to. We can just work from these tables when using multivector geometric multiplication.
Below are the tables for various grades of multivector
Geometric Multiplication for 2D multivectors.
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 |
Geometric Multiplication for 3D multivectors.
a*b |
b.e | b.e1 | b.e2 | b.e3 | b.e12 | b.e31 | b.e23 | b.e123 |
| a.e | 1 | e1 | e2 | e3 | e12 | e31 | e23 | e123 |
| a.e1 | e1 | 1 | e12 | -e31 | e2 | -e3 | e123 | e23 |
| a.e2 | e2 | -e12 | 1 | e23 | -e1 | e123 | e3 | e31 |
| a.e3 | e3 | e31 | -e23 | 1 | e123 | e1 | -e2 | e12 |
| a.e12 | e12 | -e2 | e1 | e123 | -1 | e23 | -e31 | -e3 |
| a.e31 | e31 | e3 | e123 | -e1 | -e23 | -1 | e12 | -e2 |
| a.e23 | e23 | e123 | -e3 | e2 | e31 | -e12 | -1 | -e1 |
| a.e123 | e123 | e23 | e31 | e12 | -e3 | -e2 | -e1 | -1 |
