Dot Product Associates with Scalar Multiplication/Proof 3

Theorem

$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$

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.

$\blacksquare$