Dot Product of Perpendicular Vectors

Theorem
Let $\mathbf a$ and $\mathbf b$ be vector quantities.

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

Then:
 * $\mathbf a \cdot \mathbf b = 0$


 * $\mathbf a$ and $\mathbf b$ are perpendicular.

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 $\mathbf b$ be perpendicular, by definition $\theta = 90 \degrees$.

The result follows by Cosine of Right Angle, which gives that $\cos 90 \degrees = 0$.