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What is the formula of triple scalar product and how it is derived?
Dear student,
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).
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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
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