I wanted to give a beginning student of vector calculus a deeper understanding of dot and cross products and fell into some questions for which I have no ready answers. In some sense one feels that dot and cross products exhaust the possibilities of important products of vectors, and I expect there is a theorem here somewhere.
Suppose we have a "product" P of two vectors which is a vector. So we say that it is a bilinear map of $V \times V \to V$. Of course so far it is just an arbitrary map of pairs of basis vectors to vectors which extends by linearity to all vectors. Now assume also it is associative. We can define $P_+$ and $P_-$ to be the even and odd parts of $P$, so $P_+$ is commutative and $P_-$ is anticommutative. Can we relate $P_+$ to dot and $P_-$ to cross in $3$ dimensions or is more needed? Characterize all such functions $P_+$ and $P_-$.