Equivalence of Definitions of Dot Product

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

Definition 1 implies Definition 2
Let $\mathbf a \cdot \mathbf b$ be a dot product by definition 1.

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

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

Thus $\cdot$ is a dot product by definition 2.

Definition 2 implies Definition 1
Let $\mathbf a \cdot \mathbf b$ be a dot product by definition 2.

By Angle Between Vectors in Terms of Dot Product:
 * $\theta = \arccos \dfrac {\mathbf a \cdot \mathbf b} {\left\Vert{\mathbf a}\right\Vert \left\Vert{\mathbf b}\right\Vert}$

Thus $\cdot$ is a dot product by definition 1.