Scalar triple product

The box product $\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})$ , whose absolute value is a parallelepiped's volume.

The scalar triple product $\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})$ combines the cross and dot products in one expression to find the volume of the parallelepiped three vectors span: cross two of the vectors to get a perpendicular pole whose length is the base's area, then dot with the third vector to measure the height in that pole's direction. Its value can be cycled among the three vectors without changing, $\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}) = \mathbf{v}\cdot(\mathbf{w}\times\mathbf{u})$ , but swapping just two vectors flips its sign — which is why the volume formula takes the absolute value. A scalar triple product of exactly $0$ means the three vectors are coplanar, with zero enclosed volume.

Examples

For $\mathbf{u}=\langle 1,2,0\rangle$ , $\mathbf{v}=\langle 0,1,3\rangle$ , $\mathbf{w}=\langle 2,0,1\rangle$ : $\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})=13$ , so the parallelepiped they span has volume $13$ .