Dot Product Operator is Bilinear

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

Let $c$ be a real scalar.

Then:
 * $\left({c \mathbf u + \mathbf v}\right) \cdot \mathbf w = c \left({\mathbf u \cdot \mathbf w}\right) + \left({\mathbf v \cdot \mathbf w}\right)$