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Geometric Algebra Products
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| A B = | |
| ( a b + a1 b1 + a2 b2 - a12 b12) | |
| e1 ( a b1 + a1 b - a2 b1 + a12 b2 ) | |
| e2 ( a b2 + a1 b12 + a2 b - a12 b1 ) | |
| e12 ( a b12 + a1 b2 - a2 b12 + a12 b ) | |
| B A = | |
| ( a b + a1 b1 + a2 b2 - a12 b12) | |
| e1 ( a b1 + a1 b + a2 b1 - a12 b2 ) | |
| e2 ( a b2 - a1 b12 + a2 b + a12 b1 ) | |
| e12 ( a b12 - a1 b2 + a2 b12 + a12 b ) | |
| A B + B A = | |
| (2 a b + 2 a1 b1 + 2 a2 b2 - 2 a12 b12) | |
| e1 (2 a b1 + 2 a1 b + 0 a2 b1 - 0 a12 b2 ) | |
| e2 (2 a b2 - 0 a1 b12 + 2 a2 b + 0 a12 b1 ) | |
| e12 (2 a b12 - 0 a1 b2 + 0 a2 b12 + 2 a12 b ) | |
| A B - B A = | |
| (0 a b + 0 a1 b1 + 0 a2 b2 - 0 a12 b12) | |
| e1 (0 a b1 + 0 a1 b - 2 a2 b1 + 2 a12 b2 ) | |
| e2 (0 a b2 + 2 a1 b12 + 0 a2 b - 2 a12 b1 ) | |
| e12 (0 a b12 + 2 a1 b2 - 2 a2 b12 + 0 a12 b ) | |
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The solution seems to be following:
The above calculations are true, but as a convention, the outer product is extended to work for with scalars.