Equivalence of Definitions of Dot Product

Theorem
Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^n$.

General Context implies Definition by Cosine
Let $\mathbf a \cdot \mathbf b$ be a dot product in its general context.

From Cosine Formula for Dot Product:
 * $\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$

where $\theta$ is the angle between $\mathbf v$ and $\mathbf w$.

Thus $\cdot$ is a dot product by cosine definition.

Definition by Cosine implies General Context
Let $\mathbf a \cdot \mathbf b$ be a dot product by cosine definition.

Let $\mathbf a$ and $\mathbf b$ be expressed in the form:

Then we have:

Thus $\cdot$ is a dot product in its general context.