Scalar Triple Product Calculator

Summary

Enter three 3D vectors to compute a · (b × c) and find the signed volume of the parallelepiped they form.

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