Well, I could imagine the reason for the latter - due to the convention to write the cross product as $\alpha_1 \times \alpha_2 \times \dots \times \alpha_n$.
But for the others - where do their names come from?
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Note that since the cross product is not associative, writing $\alpha_1\times\alpha_2\times\cdots\times\alpha_n$ is really not a good idea. – hmakholm left over Monica May 22 '16 at 19:03
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Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics is a good resource for this sort of thing. – May 22 '16 at 19:15
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@HenningMakholm that formula makes sense iff $\alpha_i \in \mathbb{R}^{n+1}$ – marmistrz May 23 '16 at 15:40
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Tensor products trace their name back back to continuum mechanics, which concerns, among other things, how elastic objects react to being squeezed, stretched, twisted, sheared and so forth -- in other words, the study of tension inside materials.
This theory contained a number of quantities that come in sets indexed by two or more coordinate directions, with complex rules for how to combine them and how to re-express them to describe the same physical situation seen from a different (e.g. rotated) coordinate system. It was realized that several of those quantities followed similar rules, and things that follow those rules became known as tensors. You can google various kinds of "stress tensor" that are still in use.
In mathematics, the rules for tensors became formalized as multilinear algebra, first for finite-dimensional vector spaces (because that's what physics needs) and then generalized to abstract vector spaces, and further yet to tensor products of general modules.
Then category theory got in on the game generalized it to absurd degrees; and certain parts of category theory fell in love with the $\otimes$ symbol -- to the extent that the terminology of monoidal categories use "tensor product" for just about any way to create one space/structure out of two in a halfway well-behaved fashion.
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