Proof of the geometric formula for the dot product

Proof

We argue directly from the definition of the angle.

1. Let $u$ and $v$ be nonzero vectors in $\R^n$, and let $\theta$ be the angle between them.
2. Since $u \neq 0$, some coordinate $u_i$ is nonzero. So the length $\norm{u} = \sqrt{u_1^2 + \dots + u_n^2}$ is positive. In the same way $\norm{v} > 0$.
3. By the definition of the angle, $\theta$ is the number in $[0, \pi]$ with $\cos \theta = \dfrac{u \cdot v}{\norm{u} \, \norm{v}}$. Here $u \cdot v$ is the dot product.
4. The denominator $\norm{u} \, \norm{v}$ is positive by step 2. We multiply both sides of the equation in step 3 by it.
5. This gives $u \cdot v = \norm{u} \, \norm{v} \cos \theta$, which is the statement.