Scalar Triple Product & Parallelepiped Volume

Understanding the scalar triple product calculator — concept and principle

The scalar triple product a · (b × c) combines three 3D vectors into a single number. Its absolute value is the volume of the parallelepiped (a slanted box) whose edges are a, b and c, and its sign tells you whether the three vectors form a right-handed or left-handed set. This calculator returns the triple product, the parallelepiped volume, the volume of the tetrahedron with the same three edges, and whether the vectors are coplanar. It is used in solid geometry, physics, crystallography (unit-cell volume) and 3D graphics.

How it is calculated

• a · (b × c) equals the determinant of the 3×3 matrix whose rows are a, b and c:
• det = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁).
• Parallelepiped volume = |a · (b × c)|.
• Tetrahedron volume = |a · (b × c)| ÷ 6.
• Coplanar if the triple product is 0 (within 10⁻¹²).

The principle

The vector b × c is perpendicular to the base parallelogram spanned by b and c, and its length equals the area of that base. Taking the dot product with a multiplies this area by the component of a along the normal, which is the height of the box. Base × height is the volume. If a lies in the plane of b and c, the height is zero, so the volume and the triple product vanish. A tetrahedron with the same three edges fills exactly one sixth of the parallelepiped. Cyclic swaps keep the value, a · (b × c) = b · (c × a) = c · (a × b), while swapping two vectors flips the sign.

Worked example

1. Defaults: a = (1, 0, 0), b = (0, 2, 0), c = (1, 1, 3).
2. b × c = (2·3 − 0·1, 0·1 − 0·3, 0·1 − 2·1) = (6, 0, −2).
3. a · (b × c) = 1·6 + 0·0 + 0·(−2) = 6.
4. Parallelepiped volume = |6| = 6.
5. Tetrahedron volume = 6 ÷ 6 = 1.
6. The product is not 0, so coplanar: no.

How to read the result

A positive value means a, b, c are right-handed (like the x, y and z axes); negative means left-handed. The volume is in cubic units of whatever length unit the components use. A result of zero means the vectors lie in one plane, so the matrix [a; b; c] is singular and the vectors are linearly dependent. In the example, b and c span a base of area |b × c| = √40 ≈ 6.32, and a contributes a height of 6 ÷ √40 ≈ 0.949.

Common mistakes

• Computing (a · b) × c, which is not defined; the cross product must be done first.
• Forgetting the absolute value when a volume is wanted.
• Using the tetrahedron formula with edges that do not share a common vertex.

Frequently asked questions

How do I find the volume of a tetrahedron from its four vertices?

Take one vertex P and form a = Q − P, b = R − P, c = S − P. The volume is |a · (b × c)| ÷ 6.

Does the order of the vectors matter?

Only for the sign. Cyclic orders give the same value; swapping two vectors changes the sign.

Is it the same as the box product?

Yes, [a, b, c] is another common notation for the scalar triple product.

Sources: Gilbert Strang, Introduction to Linear Algebra, chapter on determinants (volume interpretation) · James Stewart, Calculus: Early Transcendentals, section on the cross product (triple products)