What is the relationship between vec a,vec b,vec c if (vec…

What is the relationship between $\vec a,\vec b,\vec c$ if $(\vec a\times\vec b)\cdot\vec c=0$?

Step 1: Geometric meaning

$|(\vec a\times\vec b)\cdot\vec c|$ is the volume of the parallelepiped on $\vec a,\vec b,\vec c$. The sign is $+$ for a right-handed triple and $-$ for a left-handed one.

Step 2: Cases

\[ (\vec a\times\vec b)\cdot\vec c=|\vec a\times\vec b|\,|\vec c|\cos\psi=0 \]

This happens exactly when one of the following holds:

• $\vec a\times\vec b=\vec 0$: $\vec a$ and $\vec b$ are collinear (or one is zero);
• $\vec c=\vec 0$;
• $\psi=90^\circ$: $\vec c$ is perpendicular to the normal $\vec a\times\vec b$, i.e. $\vec c$ lies in the plane of $\vec a,\vec b$.

The first two cases are degenerate forms of the third.

\[(\vec a\times\vec b)\cdot\vec c=0\iff\vec a,\vec b,\vec c\ \text{are coplanar (linearly dependent)}\]

Algebraically, the determinant of the coordinates is zero exactly when its rows are linearly dependent.