Maths - Bivectors

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:

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.


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