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