HomeCalculatorsMathGeometryVector Normalize Calculator
Math calculator

Vector Normalize Calculator

Unit vector û = u / |u|. Same engine as Vector. uz defaults to 0. Zero vector is invalid. Not /vector-magnitude. 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 · v1.6.0
Input interpretation
Enter values to calculate.
Result
—
Model
Unit vector û = u / |u|.
Scope
Euclidean R²/R³
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
Specification basis
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Embedded snapshot: unpublished · Build schema 1.6.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 161 remain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.

Unit vectorû = u / |u|
Magnitude|u| = √(ux² + uy² + uz²)
iSame engine as /vector. Default (3, 4, 0) → (0.6, 0.8, 0). Not a second engine.

How to use

1

Enter ux and uy

uz is optional and defaults to 0. (3, 4) normalizes to (0.6, 0.8).

2

Read û

The result has length 1. The zero vector is invalid.

Example calculations

Common configurations with formula and result.

ϟ

3-4

u = (3, 4, 0)

(3, 4) / 5
(0.6, 0.8, 0)
ϟ

Unit i

u = (1, 0, 0)

û = i
(1, 0, 0)

Vector Normalize calculator specification

Version 1.6.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
û = u / |u| with |u| = √(ux²+uy²+uz²). This page is mode=normalize on math.vector. uz defaults to 0. The zero vector has no direction. Not /vector-magnitude. Not a second engine. Not CAS.
What it calculates
Unit vector û = u / |u|.
Inputs
  • ux
  • uy
  • uz?
Outputs
  • nx
  • ny
  • nz
  • magnitude
Formula
û = u / |u|
Assumptions
  • Euclidean R²/R³
  • uz defaults to 0
  • Not /vector-magnitude
  • Not a matrix
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • missing ux or uy → MISSING_REQUIRED_INPUT
  • zero vector → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
ux=3 uy=4 → (0.6, 0.8, 0)
Validation cases

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

  • ux=3 uy=4 → nx=0.6 ny=0.8 nz=0
  • ux=0 uy=0 → INVALID_INPUT
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.6.0

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.

→ 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. › → Annulus Calculator Annulus A = π(R² − r²) and P = 2π(R + r) with outer R > inner r. Same engine as Plane Geometry. Not /circle. Not a square. Not CAS. Runs locally. › → Arc Calculator Circle arc s = r θ from radius and central angle. Same engine as Circle. Not /sector. Not /spherical-sector. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Discovery /sector, /arc. Not the Pythagorean page. Not /spherical-sector. Not CAS. 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. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second vector engine?

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

Is this the magnitude page?

No. |u| stays on /calc/math/vector-magnitude. This page returns the unit vector.

Where does this run?

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