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Math calculator

Vector Calculator

R²/R³ magnitude, normalize, add/sub, dot, cross, angle, projection, and scalar triple product. Discovery /vector-magnitude, /dot-product, /cross-product. 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
R²/R³ vector magnitude, normalize, add/sub, dot, cross, angle, projection, and scalar triple product.
Scope
Euclidean R²/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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

Magnitude|u| = √(ux² + uy² + uz²)
Dotu · v = ux vx + uy vy + uz vz
Crossu × v
iAngle is the vector included angle, not the acute angle between two lines. Matrices stay on /linear-system. Discovery pages lock one mode.

How to use

1

Choose an operation

Magnitude and normalize need u. Add, sub, dot, cross, angle, and projection also need v. Triple product needs w.

2

Read the scalar or the vector

(3, 4, 0) has length 5. i × j = k. i · (j × k) = 1.

Example calculations

Common configurations with formula and result.

ϟ

3-4 magnitude

u = (3, 4, 0)

√(9+16)
5
ϟ

Dot 1-2-3 · 4-5-6

u = (1, 2, 3), v = (4, 5, 6)

1·4 + 2·5 + 3·6
32
ϟ

i × j

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

k
(0, 0, 1)

Vector calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A Euclidean vector u = (ux, uy, uz) has magnitude |u| = √(ux²+uy²+uz²). Dot is u·v, cross is u×v, the included angle is arccos((u·v)/(|u||v|)) in [0°, 180°], and the scalar triple product is u·(v×w). uz defaults to 0. Not a matrix. Not coordinate geometry. Not CAS.
What it calculates
R²/R³ vector magnitude, normalize, add/sub, dot, cross, angle, projection, and scalar triple product.
Inputs
  • mode
  • ux
  • uy
  • uz?
  • vx?
  • vy?
  • vz?
  • wx?
  • wy?
  • wz?
Outputs
  • magnitude
  • nx
  • ny
  • nz
  • dot
  • cx
  • cy
  • cz
  • angle_deg
  • triple
Formula
|u| = √(ux² + uy² + uz²)
Assumptions
  • Euclidean R²/R³
  • uz/vz/wz default to 0
  • Not a matrix
  • Not coordinate geometry
  • Not CAS
Units
  • length (consistent units)
  • degrees
Boundary conditions
  • missing ux or uy → MISSING_REQUIRED_INPUT
  • zero vector normalize / angle / projection → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
magnitude (3,4,0) → 5
Validation cases

3 published on this page · 11/11 tests · Production surface contract 3/3 · View evidence

  • mode=magnitude ux=3 uy=4 → magnitude=5
  • mode=dot ux=1 uy=2 uz=3 vx=4 vy=5 vz=6 → dot=32
  • mode=cross ux=1 uy=0 uz=0 vx=0 vy=1 vz=0 → cz=1
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, Cross Product Calculator, Cylinder Calculator, Dot Product 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. › → 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. › → 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. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a matrix calculator?

No. Det, inverse, eigen, and subspaces stay on /calc/math/linear-system.

Is the angle the same as angle-between-lines?

No. This is the included angle of two direction vectors in [0°, 180°]. The acute angle between two lines is /calc/math/angle-between-lines.

Where does this run?

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