The scalar triple product a·(b×c) combines the cross product and the dot product into a single number that measures how three vectors relate in 3D space. It tells you the signed volume of the parallelepiped the vectors span, and whether the three vectors are coplanar. This calculator is built for students and engineers who need a quick, reliable answer without expanding the determinant by hand.
How the scalar triple product works
The scalar triple product is computed as a · (b × c): first take the cross product of b and c, which produces a new vector perpendicular to both, then take the dot product of that result with a. Algebraically, this is identical to stacking a, b, and c as the three rows of a 3×3 matrix and computing its determinant — this calculator uses the determinant form internally because it avoids intermediate rounding from computing the cross product as a separate step. The result is always a single number, never a vector.
Inputs and what they mean
Every field is a component of one of three 3D vectors: ax, ay, az for vector a; bx, by, bz for vector b; and cx, cy, cz for vector c. All nine fields are required — there is no 2D mode, since the scalar triple product is only defined for three vectors in three dimensions. Order matters: a·(b×c) is the same as b·(c×a) and c·(a×b) (the cyclic property), but swapping any two vectors, or swapping any two components of the same vector inconsistently, flips the sign of the result.
Limits and edge cases
When the result is exactly 0, the three vectors are coplanar — they lie in a single plane and span zero volume. This also happens if any two of the three vectors are identical or parallel, or if any vector is the zero vector (0, 0, 0). Because floating-point arithmetic can leave tiny rounding noise (for example 0.0000000000000002 instead of a clean 0), this calculator snaps near-zero results to exactly 0 using a scale-aware tolerance, so genuinely coplanar inputs are reported as coplanar rather than as a stray near-zero number.