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

(a × b) ⋅ c = | | || - 1 3 3 || | - 2 3 1 | || 0 4 0 ||
= -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.