What is the formula of triple scalar product and how it is derived?

Dear student,

The scalar triple product

The scalar triple product of three vectors a, b, and c is (a × b) ⋅ c. It is a scalar product because it evaluates to a single number . The scalar triple product is important because its absolute value |(a × b) ⋅ c| is the volume of the parallelepiped spanned by a, b, and c (i.e., the parallelepiped whose adjacent sides are the vectors a, b, and c).

This results from simple geometry. The volume is the area of the base times the height. We already know the area of the parallelogram base: ||a × b||. The height is the component of c in the direction normal to the base, i.e., in the direction of a × b. Hence the height is ||c|| | cos ?|, where ? is the angle between c and a × b. (Why do we need the absolute value? cos ? would be negative if ? > π/2.)

The volume of the parallelepiped is therefore