Properties of Dot Product

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

Let $c$ be a real scalar.

The dot product has the following properties:

Dot Product Operator is Bilinear

 * $(\mathbf u + \mathbf v) \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$


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

Proofs
From the definition of dot product


 * $\displaystyle \mathbf a \cdot \mathbf b = \sum_{i \mathop = 1}^n a_i b_i$

Alternative Proofs
Because these properties are used to demonstrate the equivalence of the definitions of the dot product, it is necessary to derive them for both definitions.

From our Alternative Definition of Dot Product
 * $\displaystyle \mathbf a \cdot \mathbf b = \left\Vert{ \mathbf a }\right\Vert \left\Vert{ \mathbf b }\right\Vert \cos \angle \mathbf a, \mathbf b$