Inputs
Example data table
Use these sample vectors to verify your setup and exports.
| Case | a | b | c | a · (b × c) | Orientation |
|---|---|---|---|---|---|
| 1 | [1, 2, 3] | [4, 5, 6] | [7, 8, 9] | 0 | Coplanar / zero volume |
| 2 | [1, 0, 0] | [0, 1, 0] | [0, 0, 1] | 1 | Right-handed |
| 3 | [2, -1, 4] | [0, 3, 5] | [-2, 1, 1] | -31 | Left-handed |
Formula used
The scalar triple product is a · (b × c). It equals the determinant of the 3×3 matrix with rows (or columns) a, b, c and represents the signed volume of the parallelepiped spanned by the vectors.
- b × c = (bycz − bzcy, bzcx − bxcz, bxcy − bycx)
- a · (b × c) = ax(bycz − bzcy) − ay(bxcz − bzcx) + az(bxcy − bycx)
The vector triple product uses the identity: a × (b × c) = b(a·c) − c(a·b). This expands a nested cross product into dot products and scaled vectors.
How to use this calculator
- Enter the x, y, z components for vectors a, b, and c.
- Choose a mode to emphasize either the scalar or vector triple product.
- Set decimals for display and exports; optionally add a units label.
- Press Calculate to view results above the form.
- Use Download CSV or Download PDF after a calculation.
Tip: If a · (b × c) is zero, the vectors are coplanar.
Technical article
1) What triple products measure
Triple products compress three vector directions into one quantity that highlights 3D geometry. In physics, they often appear when a quantity depends on “how much” three directions fail to be coplanar, such as volumes, orientations, and coupled rotational effects.
2) Scalar triple product as volume
The scalar triple product a · (b × c) equals the signed volume of the parallelepiped spanned by a, b, and c. A value of 1 for unit basis vectors indicates a unit volume. This calculator reports both signed and absolute volume for direct physical interpretation.
3) Determinant interpretation
The same result is obtained from the 3×3 determinant formed by stacking the vectors as rows (or columns). Determinants scale linearly with each vector, so doubling a doubles the triple product, and swapping two vectors flips the sign. This makes the value useful for consistency checks in computations.
4) Orientation and handedness
The sign indicates handedness: positive values correspond to right-handed orientation and negative values to left-handed. For the sample case a=[2,-1,4], b=[0,3,5], c=[-2,1,1], the scalar triple product is -31, confirming a left-handed ordering.
5) Vector triple product identity
The vector triple product a × (b × c) is a vector, not a scalar. Using the identity a×(b×c)=b(a·c)−c(a·b) avoids two cross products and is often numerically cleaner. The calculator reports this expanded result alongside b×c.
6) Physics applications
Triple products show up in rigid-body dynamics (moments and coupled rotations), electromagnetism (vector identities in field manipulations), fluid mechanics (vorticity-related simplifications), and crystallography (cell volumes from lattice vectors). They also help verify orthogonality and basis quality in simulations and coordinate transforms.
7) Numerical stability and units
Units follow your vector inputs: if vectors are lengths, the scalar triple product carries cubic length units. Very large or tiny components may lead to floating-point rounding; use the decimals control for readable output, and compare the determinant and dot-cross forms as a quick sanity check.
8) Interpreting zero and near-zero results
Exactly zero means the vectors are coplanar and the spanned volume is zero. Near-zero values can indicate near-coplanarity or numerical cancellation. In experimental workflows, this often flags alignment issues, weak independent directions, or an ill-conditioned basis.
FAQs
1) What does a · (b × c) represent?
It is the signed volume of the parallelepiped formed by vectors a, b, and c. The magnitude gives volume; the sign indicates right-handed or left-handed orientation.
2) Why can the result be negative?
Negative values mean the ordered set (a, b, c) is left-handed. Swapping any two vectors flips the sign, while the absolute value stays the same.
3) When is the triple product exactly zero?
When a, b, and c are coplanar or one vector is the zero vector. Geometrically, the spanned volume collapses to zero.
4) What is the vector triple product used for?
It rewrites nested cross products in dynamics and field algebra. The identity a×(b×c)=b(a·c)−c(a·b) often simplifies equations and reduces computation.
5) Do units affect the calculation?
The math is unit-agnostic, but the physical interpretation depends on units. If vectors are in meters, the scalar triple product is in cubic meters.
6) Why show both determinant and dot-cross forms?
They should match. Seeing both helps catch data-entry mistakes and provides a quick numerical consistency check, especially in automated workflows.
7) How should I handle near-zero results?
Treat them as a warning of near-coplanarity or rounding. Increase precision, rescale inputs, and verify vectors are sufficiently independent for your physical model.
Accurate triple products help reveal geometry in motion always.