Dot Product Operator is Bilinear

Theorem
Let $\mathbf u, \mathbf v, \mathbf w$ be vectors in the real Euclidean space $\R^n$.

Let $c$ be a real scalar.

Then:
 * $\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$