Scalar Triple Product Calculator

Work out scalar triple product instantly with clear inputs, formula shown and shareable results.

About Scalar Triple Product Calculator

The scalar triple product a·(b × c) equals the determinant of the matrix with these vectors as rows. Its absolute value is the volume of the parallelepiped they span, and zero means they are coplanar.

Formula

Scalar triple product

a · (b × c) = det[a; b; c], with volume = |a · (b × c)|

Worked example

With the default values below, Scalar Triple Product Calculator returns a · (b × c) of -3.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Vector a: 1, 2, 3
Vector b: 4, 5, 6
Vector c: 7, 8, 10

Results

a · (b × c): -3.00000000
Volume of the parallelepiped: 3.00000000
b × c: (2, 2, -3)
Coplanarity: Not coplanar

Step by step

1Values used
2Scalar triple product
3a · (b × c)= -3.00000000
4Volume of the parallelepiped= 3.00000000
5b × c= (2, 2, -3)
6Coplanarity= Not coplanar

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For (1,2,3), (4,5,6) and (7,8,10)?

b × c = (2, 2, −3), so the triple product is 2 + 4 − 9 = −3 and the volume is 3.

Does the order matter?

Cyclic rotations leave it unchanged; swapping any two vectors flips the sign but not the volume.