Dot Product Associates with Scalar Multiplication/Proof 3

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

Let $c$ be a real scalar.

Then:


 * $\left({c \mathbf u}\right) \cdot \mathbf v = c \left({\mathbf u \cdot \mathbf v}\right)$

Proof
From Dot Product Operator is Bilinear:
 * $\left({c \mathbf u + \mathbf v}\right) \cdot \mathbf w = c \left({\mathbf u \cdot \mathbf w}\right) + \left({\mathbf v \cdot \mathbf w}\right)$

Setting $\mathbf v = 0$ and renaming $\mathbf w$ yields the result.