I know how to compute scalar and cross products but I'm not very familiar with what other rules applies to vectors.
For example If I have the vectors u and v and the following equation and I know that u x v = {3,1,-1}
how do they arrive at 4uxv when simplifying the equation? Also in what order should you do the calculations, does cross product come before dot and addition/subtraction?
the cross-product distributes over addition, so
a x (b + c) = (a x b) + (a x c).
here, your "a" is -v, the "b" is 3u, and the "c" is v, so we have:
-v x (3u + v) = (-v x 3u) + (-v x v)
the cross-product is also compatible with scalar multiplication:
(ra) x b = a x (rb) = r(a x b).
in particular, for r = -1, and a = b = v:
-v x v = (-1)v x v = (-1)(v x v) = (-1)(0) = 0.
thus
(-v x 3u) + (-v x v) = (-v x 3u) + 0 = -v x 3u