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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 ) | |
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.
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According to Wikipedia the inner and outer products can be defined by using the definition using the geometric product.
Here you can see the geometric products of general 2D multivectors
AandBin Clifford algebra, which is equivalent to some common geometric algebra.As you may see, the outer product of any two multivectors does not contain a scalar part.
But according to most documentation, the outer product is exactly like the geometric product, just that it gets zero, when two basis vectors are the same. So multiplication of scalars would just be normal multiplication, also for the outer product. But when it has to be true, that the inner product and outer product sum to the geometric product, scalar multiplication using the inner product would be zero.
That's also not the behaviour, I would expect. When looking at implementations of geometric algebra (like versor or bld-ga, they return the same scalar value for each kind of multiplication of two scalar values. So the inner and outer products I'm looking for are not the products, which sum to the geometric product.