Scalar triple product example

Example

Find the volume of the parallelepiped spanned by the vectors a = (- 2, 3, 1), b = (0, 4, 0), and c = (- 1, 3, 3).

Solution: The volume is the absolute value of the scalar triple product of the three vectors.

The triple product is

$$\left\vert\vphantom{ \begin{array}{rrr} -1 & 3 & 3 -2 & 3 & 1 0 & 4 & 0 \end{array} }\right. \begin{array}{rrr} -1 & 3 & 3 -2 & 3 & 1 0 & 4 & 0 \end{array} \left.\vphantom{ \begin{array}{rrr} -1 & 3 & 3 -2 & 3 & 1 0 & 4 & 0 \end{array} }\right\vert$$ (a × b) · c = - 1(0 - 4) - 3(0 - 0) + 3(- 8 + 0) = 4 - 24 = - 20

Hence the volume is | - 20| = 20.

Below is the CVT used to illustrate the scalar triple product. However, here the vectors fixed at the above values.