Scalar Triple Product Calculator

Summary

The scalar triple product of three vectors a, b, and c is defined as a · (b × c). It produces a single scalar value equal to the signed volume of the parallelepiped whose edges are formed by the three vectors. If the result is zero, the three vectors are coplanar. This calculator shows each intermediate step — the cross product b × c followed by the dot product with a.

How it works

1. Enter the x, y, and z components for each of the three vectors a, b, and c.
2. The calculator first computes the cross product b × c, producing an intermediate vector.
3. It then takes the dot product of a with that intermediate vector.
4. The final scalar result equals the signed volume of the parallelepiped spanned by a, b, and c.
5. A result of zero means the three vectors are coplanar (lie in the same plane).

Use cases

• Determine the volume of a parallelepiped in 3D geometry.
• Test whether three vectors are coplanar (scalar triple product = 0).
• Check if a set of vectors forms a right-handed or left-handed system.
• Solve physics problems involving torque or angular momentum in three dimensions.
• Verify linear independence of three 3D vectors.
• Compute determinants of 3x3 matrices (the scalar triple product equals the determinant).

Frequently Asked Questions