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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, /vector-normalize, /vector-add, /vector-sub, /dot-product, /cross-product, /vector-angle, /vector-projection, /vector-triple. 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
R²/R³ vector magnitude, normalize, add/sub, dot, cross, angle, projection, and scalar triple product. R² packs get an engine-linked arrow figure.
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
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — vector algebra in R³
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.

Magnitude|u| = √(ux² + uy² + uz²)
Dotu · v = ux vx + uy vy + uz vz
Crossu × v
iAngle is the vector included angle, not the principal angle between two undirected 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 1.6.1 · 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. R² packs get an engine-linked arrow figure.
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 5/5 · 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
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Euclidean vector magnitude, inner product, and included angle
  • CRC Standard Mathematical Tables — vector algebra in R³
    Supports: u·v, u×v, proj_v u, and the scalar triple product u·(v×w)
Calculation version
1.6.1

Other calculators in this family: Cross Product Calculator, Dot Product Calculator, Scalar Triple Product Calculator, Vector Add Calculator, Vector Angle 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 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. › → 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 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 dedicated page is /calc/math/vector-angle. The principal angle between two undirected 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.