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Math calculator

Cross Product Calculator

Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 11/11 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
The vector (cross) product of two Euclidean vectors in R³.
Scope
Euclidean R³
Verification
Engine tested · 11/11 tests · Production surface contract 3/3 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 6 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Crossu × v
k-component(u × v)_z = ux vy − uy vx
iSame engine as /vector. Dot product is /dot-product. The scalar triple product u·(v×w) stays on the hub.

How to use

1

Enter u and v

uz and vz default to 0. i × j = (0, 0, 1).

2

Read the vector

Parallel vectors give the zero vector. The scalar product of the same pair is /dot-product.

Example calculations

Common configurations with formula and result.

ϟ

i × j

u = (1, 0, 0), v = (0, 1, 0)

k
(0, 0, 1)
ϟ

2-D

u = (3, 4, 0), v = (1, 0, 0)

k(0−4)
(0, 0, −4)

Cross Product calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
u × v = (uy vz − uz vy, uz vx − ux vz, ux vy − uy vx). This page is mode=cross on math.vector. i × j = k. 2-D inputs use uz = vz = 0 and return a k-component. Not a second engine. Not CAS.
What it calculates
The vector (cross) product of two Euclidean vectors in R³.
Inputs
  • ux
  • uy
  • uz?
  • vx
  • vy
  • vz?
Outputs
  • cx
  • cy
  • cz
Formula
u × v
Assumptions
  • Euclidean R³
  • uz/vz default to 0
  • Right-handed
  • Not a matrix
  • Not CAS
Units
  • length² (consistent units)
Boundary conditions
  • missing ux, uy, vx, or vy → MISSING_REQUIRED_INPUT
  • unknown mode → INVALID_MODE
Example
i × j → (0, 0, 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 → cx=0 cy=0 cz=1
  • ux=3 uy=4 vx=1 vy=0 → cz=-4
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cylinder Calculator, Dot Product Calculator, Ellipse Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Ellipse Calculator Standard ellipse x²/a² + y²/b² = 1. Same engine as Conic Section. a = b is a valid ellipse with e = 0; the circle-only page is /circle. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second vector engine?

No. It is mode=cross on math.vector, the same engine as /calc/math/vector.

Does 2-D cross return a scalar?

The engine always returns a 3-vector. In the plane uz = vz = 0, so only the z-component is nonzero.

Where does this run?

Locally in the browser by default. REST and MCP call the same vector engine.