Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors

Theorem
Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in a vector space of $3$ dimensions:

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

Then $\size {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }$ equals the volume of the parallelepiped contained by $\mathbf a, \mathbf b, \mathbf c$.

Proof
Let us construct the parallelepiped $P$ contained by $\mathbf a, \mathbf b, \mathbf c$.


 * Scalar-triple-product-parallelepiped.png

We have by Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors that:
 * $\mathbf b \times \mathbf c$ is a vector area equal to and normal to the area of the bottom face $S$ of $P$.

The dot product $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ is equal to the product of this vector area and the projection of $\mathbf a$ along $\mathbf b \times \mathbf c$.

Depending on the relative orientations of $\mathbf a$, $\mathbf b$ and $\mathbf c$, $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ may or may not be negative.

So, taking its absolute value, $\size {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }$ is the volume of the parallelepiped which has $\mathbf a$, $\mathbf b$ and $\mathbf c$ as edges.