Vectors are Coplanar iff Scalar Triple Product equals Zero
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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:
- $\mathbf a \cdot \paren {\mathbf b \times \mathbf c} = 0$
if and only if $\mathbf a$, $\mathbf b$ 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.
$\blacksquare$
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: $\S 4$