Scalar Triple Product Calculator
Scalar triple product u · (v × w), the signed parallelepiped volume. Same engine as Vector. Not the vector triple. Not a matrix. Not CAS. Runs locally.
Trust summary Engine tested · Specification checked · 11/11 tests · Production surface contract 3/3 · v1.6.0
- Input interpretation
- Enter values to calculate.
- Result
- —
- Model
- The scalar triple product u · (v × w), the signed parallelepiped volume.
- Scope
- Euclidean R³ (R² via uz=vz=wz=0)
- Verification
- Engine tested · 11/11 tests · Production surface contract 3/3 · Specification checked · v1.6.0
- Named expert review
- Optional · Not performed
- Specification basis
- ISO 80000-2:2019 Quantities and units — Mathematics
- Evidence
- 3 golden · 2 boundary · 6 property · Production surface contract 3/3 · Artifact integrity PASS
- Production
- Embedded snapshot: unpublished · Build schema 1.6.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 161 remain CURRENT) @ 2026-09-20T23:37:12.315Z
- Semantic contract
- PASS
Formulas
Core equations used by this calculator.
How to use
Enter u, v, and w
uz, vz, and wz are optional and default to 0. i · (j × k) = 1.
Read the signed volume
A positive result is a right-handed triad. Coplanar vectors give 0. The vector triple u × (v × w) is not this page.
Example calculations
Common configurations with formula and result.
i · (j × k)
u = (1, 0, 0), v = (0, 1, 0), w = (0, 0, 1)
Coplanar
u = (1, 0, 0), v = (0, 1, 0), w = (1, 1, 0)
Scalar Triple Product calculator specification
Version 1.6.0 · Engine tested
- Engine tested 11/11 tests · Production surface contract 3/3
- Named expert review Not performed
- Calculation version 1.6.0
- Definition
- u · (v × w) is the signed volume of the parallelepiped spanned by u, v, and w. This page is mode=triple on math.vector. uz/vz/wz default to 0. Not the vector triple u × (v × w). Not a second engine. Not CAS.
- What it calculates
- The scalar triple product u · (v × w), the signed parallelepiped volume.
- Inputs
- ux
- uy
- uz?
- vx
- vy
- vz?
- wx
- wy
- wz?
- Outputs
- triple
- Formula
u · (v × w)- Assumptions
- Euclidean R³ (R² via uz=vz=wz=0)
- uz/vz/wz default to 0
- Signed volume (orientation matters)
- Not the vector triple u × (v × w)
- Not a matrix
- Not CAS
- Units
- volume (consistent units)
- Boundary conditions
- missing ux, uy, vx, vy, wx, or wy → MISSING_REQUIRED_INPUT
- unknown mode → INVALID_MODE
- Example
- u=(1,0,0) v=(0,1,0) w=(0,0,1) → 1
- Validation cases
2 published on this page · 11/11 tests · Production surface contract 3/3 · View evidence
- ux=1 uy=0 uz=0 vx=0 vy=1 vz=0 wx=0 wy=0 wz=1 → triple=1
- ux=1 uy=0 vx=0 vy=1 wx=1 wy=1 → triple=0
- Specification basis
- ISO 80000-2:2019 Quantities and units — Mathematics
- Calculation version
- 1.6.0
Related tools
Other calculators in this family: Angle Between Lines Calculator, Annulus Calculator, Arc Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator . Explore all Geometry.
Frequently asked questions
Key distinctions behind the calculation.
Is this a second vector engine?
No. It is mode=triple on math.vector, the same engine as /calc/math/vector.
Is this the vector triple product?
No. This page returns the scalar u · (v × w). The vector triple u × (v × w) is not this mode.
What does 0 mean?
The three vectors are coplanar (or at least one is zero). The parallelepiped has zero volume.
Where does this run?
Locally in the browser by default. REST and MCP call the same vector engine.