Math calculator

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.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes

Input interpretation: Enter values to calculate.
Result: —
Verified scope: all published modes
Assurance: Core
Declared partition coverage: PASS · 10/10 declared partitions (magnitude, normalize, add, sub, dot, cross, angle, projection, triple, invalid-domain) · Matrix
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: The scalar triple product u · (v × w), the signed parallelepiped volume.
Scope: Euclidean R³ (R² via uz=vz=wz=0)
Verification: Engine tested · Source checked · v1.6.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions: Calculation 1.6.1 · CVP protocol 1.0.0-proposed · Evidence 2026-09-29.cross-product-hardening-iface2
Verification revision: 2026-09-29.cross-product-hardening-iface2 · 7/7 property · digest 3e840b22dc12
Legacy regression: 11/11 tests · Production surface contract 5/5
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 10/10 golden · 4/4 CVP boundary · 9/9 invalid · 7/7 property · 6/6 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Sources: ISO 80000-2:2019 Quantities and units — Mathematics
Gilbert Strang, Calculus — Chapter 11: Vectors and Matrices (MIT OpenCourseWare) — §§11.1–11.3 Vectors, Dot Products, Projections and Cross Products
Sources
Evidence: 3 legacy golden · 2 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 9/9 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Scalar tripleu · (v × w)

Determinantdet[u v w]

iSame engine as /vector. Default i · (j × k) → 1. Coplanar vectors → 0. Not a second engine.

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 1.6.1 · Engine tested

Calculation status

Engine tested 11/11 tests · Production surface contract 5/5
Named expert review Not performed
Calculation version 1.6.1

Review policy · Evidence

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 5/5 · 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
Sources: ISO 80000-2:2019 Quantities and units — Mathematics
Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
Gilbert Strang, Calculus — Chapter 11: Vectors and Matrices (MIT OpenCourseWare) — §§11.1–11.3 Vectors, Dot Products, Projections and Cross Products
Supports: Vector components and magnitudes, dot product and angle/projection relations, and cross product/determinant orientation and area. Projecting a point displacement onto the unit normal to Ax+By+C=0 gives distance |Ax₀+By₀+C|/sqrt(A²+B²); this planar expression is derived from the projection formula. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version: 1.6.1

Background

Interpretation and common distinctions.

What does this scalar triple product calculator compute?

u · (v × w) is the signed volume of the parallelepiped spanned by u, v, and w.

The published formula or method is u · (v × w). The following references support the stated mathematics or method:

Gilbert Strang, Calculus — Chapter 11: Vectors and Matrices (MIT OpenCourseWare): Vector components and magnitudes, dot product and angle/projection relations, and cross product/determinant orientation and area. Projecting a point displacement onto the unit normal to Ax+By+C=0 gives distance |Ax₀+By₀+C|/sqrt(A²+B²); this planar expression is derived from the projection formula. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses u = (1, 0, 0), v = (0, 1, 0), w = (0, 0, 1) with 1. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.vector v1.6.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Cross Product Calculator, Dot Product Calculator, Vector Add Calculator, Vector Angle Calculator, Vector Calculator, Vector Magnitude Calculator, Vector Normalize Calculator, Vector Projection Calculator . Explore all Vectors.

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.