Dot Product Distributes over Addition/Proof 1

Theorem

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

$\ds \left({\mathbf u + \mathbf v}\right) \cdot \mathbf w$

$=$

$\ds \sum_{i \mathop = 1}^n \left({u_i + v_i}\right) w_i$

Definition of Vector Sum and Definition of Dot Product

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^n \left({u_i w_i + v_i w_i}\right)$

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^n u_i w_i + \sum_{i \mathop = 1}^n v_i w_i$

$\ds $

$=$

$\ds \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$

Definition of Dot Product

$\blacksquare$