triple scalar product

The triple scalar product of $\vec{u}$, $\vec{v}$, and $\vec{w}$ is the determinant of the vertically concatenated matrix:

$$\left|\begin{array}{ccc}{u}_x& u_y& v_z\\ v_x& v_y& v_z\\ w_x& w_y& v_z\end{array}\right|$$

or in short:

$$\left|\begin{array}{c}\vec{u}\\ \vec{v}\\ \vec{w}\end{array}\right|$$

The above is equivalent to $\vec{u}\cdot (\vec{v}\times \vec{w})$

Triple scalar product of zero

The triple scalar product of three vectors is 0 if they are in the same plane.