I'm just wondering which takes precedence or if it really matters. It would matter wouldn't it?
For example, this is written in my textbook:
Equation for magnetic field of a point charge
so the [qv X r] in the numerator is the question I have in mind.
Here's my algebraic proof:
q = q; v = [a, b, c]; r = [d, e, f];
so if we do the cross product first we get:
(bf - ec)i + (cd - af)j +(ae - bd)k
and then distributing the scalar, it would be:
(bfq - ecq)i + (cdq - afq)j +(aeq - bdq)k
In contrast, if we do multiplication first, qv is now [aq, bq, cq]. If we cross product those two, we get the same thing.
(bfq - ecq)i + (cdq - afq)j +(aeq - bbq)k
Algebraically, it doesn't seem to matter. Are there cases where it will matter?
It's true that one of the algebraic properties of cross products is that they're compatible with scalar multiplication so that for any scalar $c \in \mathbb R$ and for any vectors $\vec u, \vec v \in \mathbb R^3$, we have that: $$ c\vec u \times \vec v = c(\vec u \times \vec v) = \vec u \times c \vec v $$