TITLE: What is "exterior" about an exterior product?
QUESTION [8 upvotes]: This is a question about terminology. What is "inner" about an inner product, or "outer" about an outer product?

REPLY [8 votes]: 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".