Is Cross Product Defined on Vector Space?

Question

In Wikipedia, a cross product between two "vectors" is defined in terms of the angle between the vectors and their magnitudes.

1. As I learned cross product in linear algebra, which I understand to be a topic about vector space, I now wonder if cross product is not defined on vector space, but instead can only be defined on an inner product space so that the angle between the vectors and their magnitudes can make sense?
2. Or is there other definition of cross product on vector space?

Thanks and regards!

Answer 1

One point of view is that the cross product is the composition of the Hodge dual and the exterior product $V \times V \to \Lambda^2 V$ in three dimensions. The Hodge dual requires additional structure to define: you need not only an inner product, but an orientation. This reflects the fact that there is a choice of handedness in the definition of the cross product.

Another point of view is that the cross product is not an operation on spatial vectors in $\mathbb{R}^3$, but the Lie bracket on the Lie algebra of $\text{SO}(3)$. This is the definition, for example, which is relevant to the physics of angular momentum. Of course, if you insist on taking the cross product of spatial vectors you need a way to identify spatial vectors with elements of the Lie algebra of $\text{SO}(3)$. Writing $\text{SO}(3)$ as $\text{Aut}(V)$ where $V$ is a real oriented 3-dimensional inner product space, the Lie algebra of $\text{SO}(3)$ is naturally isomorphic to $\Lambda^2 V$, so this identification is again the Hodge dual.

Answer 2

The question was studied by Grassmann. At the time, people laughed at him, but now you can read all about it in the Wikipedia article on Exterior Algebra.

Answer 3

You are correct in that you need an inner product to define the cross product. A geometric picture of it is as follows: take two vectors in three dimensions. They determine a parallelogram and the cross product is defined to be the vector perpendicular (with respect to the inner product) to the parallelogram and with magnitude equal to its area.

Note that there is something special in 3-dimensions that allows this definition to work, namely that to any plane there is a unique perpendicular direction. This is not the case in other dimensions so this definition does not generalize. Indeed, as the wikipedia article states, the only other dimension that has an analogous cross product is 7 (dimension 1 also has a cross product but it is trivial-- just negative of the regular product of real numbers). What is special about dimensions 1,3 and 7 is that there are division algebras only in dimensions 2,4, and 8 (the real numbers, the quaternions, and the octonions). The wikipedia article talks more about this construction.