Vectors are Coplanar iff Scalar Triple Product equals Zero

Theorem
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:

Let $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ denote the scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$.

Then:
 * $\mathbf a \cdot \paren {\mathbf b \times \mathbf c} = 0$

$\mathbf a$, $\mathbf b$ and $\mathbf c$ are coplanar.

Proof
From Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors:
 * $\size {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }$ equals the volume of the parallelepiped contained by $\mathbf a, \mathbf b, \mathbf c$.

The result follows.