Dot Product of Like Vectors

Theorem
Let $\mathbf a$ and $\mathbf b$ be vector quantities such that $\mathbf a$ and $\mathbf b$ are like.

Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ and $\mathbf b$.

Then:
 * $\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b}$

where $\norm {\, \cdot \,}$ denotes the magnitude of a vector.

Proof
By definition of dot product:
 * $\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$

where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$.

When $\mathbf a$ and $\mathbf b$ are like, by definition $\theta = 0$.

The result follows by Cosine of Zero is One, which gives that $\cos 0 \degrees = 1$.