Cosine Formula for Dot Product/Proof 2

Proof
Let $\mathbf v$ and $\mathbf w$ be considered to be embedded in a Cartesian plane $\CC$.

By Dot Product is Invariant under Coordinate Rotation, we may rotate $\CC$ arbitrarily, and $\mathbf v \cdot \mathbf w$ will not change.

So, let us rotate $\CC$ to $\CC'$ such that the $x$-axis is parallel to $\mathbf v$.

Hence $\mathbf v$ can be expressed as:

Hence: by definition of dot product:

Hence the result.