I know how to prove that that to get the dot product of two vectors (in 2-D space) you can either simply add the products of their parallel components, or multiply their magnitudes to the cosine between both vectors. Namely, I can prove these two ways of doing the dot-product are equivalent.
But how do I prove this is true in 3-D?
The way I prove this in 2-D is basically shown here: http://mathworld.wolfram.com/DotProduct.html
A simple way to do it would be to show that, if $U$ is a unitary matrix, then $\langle Ux, Uy \rangle = \langle x,y \rangle$. Indeed, if $\vec u$, $\vec v$, and $\vec w$ are mutually perpendicular vectors of length 1 which form the columns of such a matrix, then \begin{align*} \langle Ux, Uy \rangle &= \langle x_1 \vec u + x_2 \vec v + x_3 \vec w, y_1 \vec u + y_2 \vec v + y_3 \vec w \rangle\\ \end{align*} which will simplify to $\langle x,y\rangle$ using the bilinearity properties of the inner product. This helps because you can use a unitary matrix to send any pair of vectors into the $xy$-plane, where the inner product corresponds directly to a two-dimensional inner product, and the angle between the vectors does not change.