The vector cross gives a bivector rather than a vector.
Its not very strange that the product of two vectors is not a vector itself, its the same principle as the dot product which produces a scalar. In fact the cross product and dot products are complimentary.
In 3 dimensions a bivector behaves like a vector therefore, in most cases, we treat the result of cross product as a vector.
However there are some advantages in keeping track as to whether such quantities are vectors or bivectors:
- In physics, it can give us clues about the nature of a quantity, bivectors represent quantities associated with planes including rotations.
- There are some small differences between the algebra of bivectors, the dot product of two bivectors is a negative number whereas the dot product of two vectors is a positive number.
- This allows the vector cross product to be generalised to the outer product which is defined for all dimensions of vectors.
- This is a good introduction to Clifford Algebra/Geometric Algebra which is a more general type of algebra and gives a lot of insights
Multiplying the bivector by a vector will toggle it back to a vector, so it is like an exclusive-or operation, multiplying the same types gives a bivector multiplying different types gives a vector:
vector × vector = bivector
bivector × bivector = bivector
vector × bivector = vector
bivector × vector = vector
Vector Cross Product
For a more general introduction to bivectors see the pages about Clifford Algebra/Geometric Algebra.