## Scalar Triple Product

Scalar triple product is a multiplication of three vectors that produces a scalar

Example
Suppose we have three vectors , and .

First we compute cross product

Then, the scalar triple product is

Geometrically, scalar triple product is equal to the volume of parallelepiped. The based area of the parallelepiped is a parallelogram whose area is equal to the absolute value of the cross product .

The interactive program below helps you to compute Scalar triple product algebraically. You input 3 vectors of the same dimension. The program output is the scalar triple product. Random example button will generate random vectors at the right format.

(a * b ) . c

## Properties

Some important properties of related to scalar triple product are

• Scalar triple product is invariant under cyclic permutation of the vectors, that is
• Scalar triple product is zero if two of the vectors lie on the same plane or are parallel to the same plane (i.e. coplanar) .
• Three vectors are linearly dependent (coplanar) if and only if their scalar triple product is zero .
• Three vectors form a basis if and only if their scalar triple product is not zero . The basis is right handed if the scalar triple product is positive and called left handed if the scalar triple product is negative .