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Math calculator

Dot Product Calculator

Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. 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
The scalar (dot) product of two Euclidean vectors.
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

Dotu · v = ux vx + uy vy + uz vz
Angleu · v = |u| |v| cos θ
iSame engine as /vector. Cross product is /cross-product. Not a matrix inner product of higher order.

How to use

1

Enter u and v

uz and vz default to 0. (1, 2, 3) · (4, 5, 6) = 32.

2

Read the scalar

i · j = 0. The cross product of the same pair is /cross-product.

Example calculations

Common configurations with formula and result.

ϟ

1-2-3 · 4-5-6

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

4+10+18
32
ϟ

Perpendicular

i · j

0
0

Dot Product calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
u · v = ux vx + uy vy + uz vz. This page is mode=dot on math.vector. Perpendicular vectors give 0. The included angle is mode=angle on /calc/math/vector. Not a second engine. Not CAS.
What it calculates
The scalar (dot) product of two Euclidean vectors.
Inputs
  • ux
  • uy
  • uz?
  • vx
  • vy
  • vz?
Outputs
  • dot
Formula
u · v = ux vx + uy vy + uz vz
Assumptions
  • Euclidean R²/R³
  • uz/vz default to 0
  • Not a matrix
  • Not CAS
Units
  • length² (consistent units)
Boundary conditions
  • missing ux, uy, vx, or vy → MISSING_REQUIRED_INPUT
  • unknown mode → INVALID_MODE
Example
u=(1,2,3) v=(4,5,6) → 32
Validation cases

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

  • ux=1 uy=2 uz=3 vx=4 vy=5 vz=6 → dot=32
  • ux=1 uy=0 vx=0 vy=1 → dot=0
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, Ellipse 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. › → Ellipse Calculator Standard ellipse x²/a² + y²/b² = 1. Same engine as Conic Section. a = b is a valid ellipse with e = 0; the circle-only page is /circle. 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=dot on math.vector, the same engine as /calc/math/vector.

Where is the cross product?

/calc/math/cross-product is mode=cross on the same engine.

Where does this run?

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