HomeCalculatorsMathGeometryVector Angle Calculator
Math calculator

Vector Angle Calculator

Included angle of two Euclidean vectors in [0°, 180°]. Same engine as Vector. Zero vectors are invalid. Not /angle-between-lines. 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
The included angle of two Euclidean vectors in [0°, 180°].
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Embedded snapshot: unpublished · Build schema 1.6.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 161 remain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.

Included angleθ = arccos((u · v) / (|u| |v|))
Range0° ≤ θ ≤ 180°
iSame engine as /vector. Default i and j → 90°. Parallel → 0°. Opposite → 180°. Not a second engine.

How to use

1

Enter u and v

uz and vz are optional and default to 0. Neither vector may be zero. i and j form 90°.

2

Read the included angle

The result is in [0°, 180°]. The acute angle between two undirected lines is /calc/math/angle-between-lines.

Example calculations

Common configurations with formula and result.

ϟ

i and j

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

arccos(0)
90°
ϟ

Parallel

u = (1, 0, 0), v = (2, 0, 0)

arccos(1)
0°

Vector Angle calculator specification

Version 1.6.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
θ = arccos((u · v) / (|u| |v|)) in [0°, 180°]. This page is mode=angle on math.vector. uz and vz default to 0. The acute angle between two lines stays on /calc/math/angle-between-lines. Not a second engine. Not CAS.
What it calculates
The included angle of two Euclidean vectors in [0°, 180°].
Inputs
  • ux
  • uy
  • uz?
  • vx
  • vy
  • vz?
Outputs
  • angle_deg
  • angle_rad
  • cosine
Formula
θ = arccos((u · v) / (|u| |v|))
Assumptions
  • Euclidean R²/R³
  • uz/vz default to 0
  • Directed vectors (not undirected lines)
  • Not /angle-between-lines
  • Not a matrix
  • Not CAS
Units
  • degree
  • radian
Boundary conditions
  • missing ux, uy, vx, or vy → MISSING_REQUIRED_INPUT
  • zero u or v → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
u=(1,0,0) v=(0,1,0) → 90°
Validation cases

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

  • ux=1 uy=0 vx=0 vy=1 → angle_deg=90
  • ux=1 uy=0 vx=2 vy=0 → angle_deg=0
  • ux=1 uy=0 vx=-1 vy=0 → angle_deg=180
  • ux=0 uy=0 vx=1 vy=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=angle on math.vector, the same engine as /calc/math/vector.

Is this the angle between two lines?

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

What if a vector is zero?

The angle is undefined. The engine returns INVALID_INPUT.

Where does this run?

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