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.