# Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($\mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($\mathbb{R}^3$) in some way?

A quick search reveals that they are, but yet the outer product of two column vectors in $\mathbb{R}^3$ is a 3x3 matrix, not another column vector. What's the link? Thanks!

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–  joriki Aug 13 '12 at 10:17
Just found this, on the ambiguity of the term "outer product": en.wikipedia.org/wiki/… –  Ryan Aug 13 '12 at 11:28

Interior product is degree descending operation. It looks like inner product, but this is just an analogy. Here is an example of how one can define interior product given some $x\in V$. You define interioir product of vectors $x_1,\ldots,x_{n-1}$ as exterior product of $x, x_1,\ldots, x_n$. Note that interioir product depends on the choice of vector $x$. Interioir products are closely related to annihilation operators on the Fock space in quantum mechanics. –  Norbert Aug 13 '12 at 11:15