Where this calculation appears
It tests coplanarity, orientation, volume, and basis handedness.
Math calculator
Scalar Triple Product Calculator
Compute a·(b×c) and its associated 3D volume. The displayed scalar triple product includes enough working to inspect signs and scale.
Scalar Triple Product inputs
Numerical setup
A numerical walkthrough
The sample product is 6, so the parallelepiped volume is 6 cubic units.
What the formula is measuring
The scalar triple product is an oriented parallelepiped volume and equals the determinant formed by three 3D vectors.
The roles assigned to vector a, vector b and vector c explain the operation that produces scalar triple product.
Conditions that affect Scalar Triple Product
Swapping two vectors reverses the sign, while absolute value gives unsigned volume. If the Scalar Triple Product assumptions do not fit, consider intermediate normal.
When scalar triple product looks surprising, restore the sample and vary vector c by itself.
Manual method
Compute b×c, dot with a, and verify that coplanar vectors produce zero. Scalar Triple Product also connects to determinant interpretation.
What changes—and what does not
Write each source value under its matching label before calculating: vector a, vector b and vector c. This preserves the assumptions behind scalar triple product and makes a later check possible without reopening the original problem.
Scaling one vector scales the result by the same factor. This relationship remains useful even when the final scalar triple product is rounded.
Before publishing or sharing scalar triple product, decide on units, significant digits, and whether an exact form is expected. Those choices belong to interpretation rather than the calculator engine.
Cross product supplies the intermediate normal vector. The two results may share inputs while retaining different meanings.
When reporting the answer, state the scalar triple product first, then its value and unit. Add vector a, vector b and vector c if someone else must verify the work independently.
Checking orientation and volume
The calculation behind Scalar Triple Product starts from Vector a, Vector b, and Vector c. A cyclic permutation keeps the value, while swapping any two vectors reverses its sign. Its absolute value is the parallelepiped volume.
For Scalar Triple Product, save a dot-product, magnitude, or perpendicularity check that can reproduce the geometry.
Questions about Scalar Triple Product
Checking What does Scalar Triple Product calculate?
The scalar triple product is an oriented parallelepiped volume and equals the determinant formed by three 3D vectors.
When is Scalar Triple Product useful?
It tests coplanarity, orientation, volume, and basis handedness.
What can make Scalar Triple Product misleading?
Swapping two vectors reverses the sign, while absolute value gives unsigned volume.
Scalar Triple Product independently
Compute b×c, dot with a, and verify that coplanar vectors produce zero.