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 $\mathbf a, \mathbf b, \mathbf c$.
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.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $7$. Products of Three Vectors
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 4$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Miscellaneous Formulas involving Dot and Cross Products: $22.17$