How can the vector distributive law be proven?

Question

i.e. a x (y + z) = a x y + a x z, where a, y and z are vectors? If possible I'd prefer an intuitive argument, such as a geometric representation, but a rigorous proof would also be helpful.

Answer 1

it will never be proven as it is only a theory.

Answer 2

It follows directly from the definition of the dot product and the distributive law of scalars:
http://www.math.psu.edu/levi/230.notes/HW1.pdf

A graphical proof of distrubutivity for the cross-product is shown here:
http://web.mit.edu/6.013_book/www/appendices/app1.html

.