TITLE: Why is the × operator not defined for vector × vector?
QUESTION [5 upvotes]: For a vector space, the + operator maps two vectors to another vector while the × operator maps a scalar and a vector to another vector. 
To me, it seems strange that scalars are seen as separate to vectors when defining a vector space, with the x operator being specially created to map a scalar and vector to vectors.
Why can't instead the × operator map two vectors to another vector while remaining consistent with the scalar × vector operation?
For example, $(a, b, c) × (d, e, f) = (ad, ae, af) + (bd, be, bf) + (cd, ce, cf)$
where $ad$ scales $d$ by a factor of $a$; $bd$ rotates $d$ by an amount $b$ about some axis, $cd$ rotates $d$ about another axis by $c$.
This way there would be no need to bring in an additional set of scalars, providing it was consistent.

REPLY [8 votes]: The concept of a vector space isn't so much defined as recognized. All over Mathematics there are sets of things that have a natural addition, and a natural multiplication by real or complex numbers, but no natural way to multiply two things in the set to get another thing in the set. Once you've seen enough of these things, you accept that that's what Mathematics is giving you, and you make your definitions accordingly. 
The definitions commonly in use lead to a huge number of useful concepts and results. Linear combination, span, linear dependence/independence, basis, dimension for starters - does your proposed multiplication lead to any of those concepts, or to any other concepts even half as useful?