I do not understand this question/series of questions:
how would I go about defining scalar multiplication in the context of 3-vectors with Cartesian coordinates and 3 x 3 matrices?
Umm, if i had to guess, I'd say the problem wants you to find a way to arrange 3 element vectors and 3x3 matrices to perform a simple scalar multiplication.
Let's say you want to multiply some scalars a and b:
a*b = [a 0 0] * [1 0 0; 0 0 0; 0 0 0] * [b; 0; 0]
That's a way to write a scalar multiplication using 3-element vectors and a 3x3 matrix..assuming that's what the question is asking.
If a is a scalar and [x, y, z] is a 3-vector then their scalar product is [ax, ay, az].
If a is a scalar and is a 3 by 3 matrix then their scalar product is