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This is a question about terminology. What is "inner" about an inner product, or "outer" about an outer product?

This terminology (or rather its literal German translation) was introduced by Grassmann.

I named the former product exterior, the latter interior, reflecting that the former was nonzero only when involving independent directions, the latter only when involving a shared, i.e., partly common one.

H. Grassmann, Die lineale Ausdehnungslehre [archive.org] (1844) x-xi.

NB the words in the original text translated in the above excerpt to exterior and interior are respectively äussere and innere, but they can also be translated respectively to outer and inner, and now all four terms have distinct meanings in this context.

For more detail, see the answer to essentially the same question on MathOverflow, from where the above translation was lifted (this borrowing motivated setting this answer as a community wiki answer):

Etymology of "exterior" in "exterior calculus".

Consider the case n=3. The interior product of two vectors is non-zero when one lives in the span of the other. The outer product of two vectors is non-zero when one lives outside the span of the other.

This terminology can be traced back to one of the earlier German texts on linear algebra but I can't quite recall the name. I'll come back if I recall it.

EDIT: Found my source. Take a look at section 1.2 of this.

The exterior (= outer) product takes values in a "higher" dimensional space: if $V$ has dimension $n$ and $v, w in V$ then $v wedge w in Lambda^2(V)$ and that space has dimension $binomn2$, which is bigger than $n$ when $n > 3$. In Euclidean space, the wedge product of two vectors is represented by a parallelogram with the original vectors as its edges.

The inner/outer terminology goes back to Grassmann.

Well if I take the inner product of two vectors I go from the full vector space down to the base field over which it's defined so its internal in that regard. As for the outer product I think of this in terms of matrix multiplication, if we reverse the inner product $x^Ty$ to $xy^T$ for column vectors in a space of dimension $n$ instead we end up with a matrix which is a larger vector space than the original space with $n^2$ dimensions. Metaphorically I think of this as identifying a vector space with a subspace of a larger space, and the rest of that space is "outer space" in the same way the Earth is in a much larger universe.