Dot Product of Vector with Itself

Theorem
Let $\mathbf u$ be a vector in the real vector space $\R^n$.

Then:
 * $\mathbf u \cdot \mathbf u = \left\Vert{\mathbf u}\right\Vert^2$

where $\left\Vert{\mathbf u}\right\Vert$ is the length of $\mathbf u$.