Dot Product Distributes over Addition

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

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

Proof 2
The following proof is valid in the context of a real Euclidean space of $3$ dimensions: