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Cross Products

I was teaching linear algebra, and I wanted a good explanation for why we can use the three-by-three determinant formula to compute it.

Lets start with some definitions. A composition algebra is one endowed with a non-degenerate quadratic form satisfying . If the base field are the reals and is positive-definite so

is an inner product, then we say it is a normed division algebra.

Theorem (Hurwitz) There are exactly four euclidean division algebras over the real numbers. Furthermore, they have dimensions .

It turns out that any higher-dimensional “cross product” that agrees with our intuition for what the standard cross product represents corresponds exactly to these four algebras, and only two of them are non-trivial.