How to prove $v_1\cdot v_2=|v_1||v_2|\cos(\theta)$ in n-dimensions?

Question

It is easy to prove in 2D that $v_1\cdot v_2=|v_1||v_2|\cos(\theta)$ where $\theta$ is the angle between $v_1$ and $v_2$.

But how to generalize? What is the proof in n-dimensions?

Answer 1

This proof uses the Law of cosines to prove it for any number of dimensions.