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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 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 included angle of two Euclidean vectors in [0°, 180°].
Scope
Euclidean R²/R³
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
Sources
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 9/9 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.6.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

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

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 principal 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 1.6.1 · 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 principal angle between two undirected 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 5/5 · 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
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 vector angle calculator compute?

θ = arccos((u · v) / (|u| |v|)) in [0°, 180°].

The published formula or method is θ = arccos((u · v) / (|u| |v|)). 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) with arccos(0). 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, Scalar Triple Product Calculator, Vector Add Calculator, Vector Calculator, Vector Magnitude Calculator, Vector Normalize Calculator, Vector Projection Calculator . Explore all Vectors.

→ 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. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → 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. › → Vector Add Calculator Componentwise vector sum u + v. Same engine as Vector. uz/vz default to 0. Not /add. Not a matrix. Not CAS. Runs locally. › → Vector Calculator R²/R³ magnitude, normalize, add/sub, dot, cross, angle, projection, and scalar triple product. Discovery /vector-magnitude, /vector-normalize, /vector-add, /vector-sub, /dot-product, /cross-product, /vector-angle, /vector-projection, /vector-triple. Not a matrix. Not CAS. Runs locally. › → Vector Magnitude Calculator Euclidean length |u| = √(ux²+uy²+uz²). Same engine as Vector. uz defaults to 0. Not a matrix. Not CAS. Runs locally. › → 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. › → Vector Projection Calculator Vector projection of u onto v. Same engine as Vector. Zero v is invalid. Not a matrix. Not CAS. Runs locally. ›

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.