The tensor product maps the vector space

into

which is a different vector space, so the notation

is misleading; it should be

(upper or lower indices depending on co- or contra variance, if such notation is relevant at all.

Whether the original vector space remains (the new space is "created" in addition) or whether the original vectors space is "destroyed" is a question of notation and context (one can destroy the "cross product" by simply changing the order or multiplication and adding).

The "outer" product is distinguished for the cross product (which relates to n=2 and the "right hand" rule) for this reason (and a number of others, including parity).

The problem the cross product raises is apparent in the problem of gimbel locking, which is why quaternions are used in action adventure games requiring rotation, among other applications.

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