Dot Product Calculator

Use our free dot product calculator to compute 𝐚·𝐛, the angle between vectors, vector projection of 𝐚 onto 𝐛, and check orthogonality.

Math

Mode

𝐚 =

𝐛 =

Formula

Compute the dot product of two vectors, the angle between them, vector projection, and orthogonality. Supports any dimension.

𝐚·𝐛 = Σ aᵢbᵢ = |𝐚||𝐛|cos(θ)

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Enter 𝐚 and 𝐛 as comma-separated numbers (same dimension).
2Select mode: dot product, projection, or orthogonality.
3Click Calculate.

Examples

Dot product (1,2,3)·(4,5,6)

1.1·4+2·5+3·6 = 32
2.cos(θ) = 32/(√14·√77) ≈ 0.9746

Result: 32

Projection of (3,4) onto (1,0)

1.scalar = 3/1 = 3
2.proj = (3,0)

Result: 3

Frequently Asked Questions

A scalar equal to Σaᵢbᵢ = |𝐚||𝐛|cos(θ), measuring how parallel vectors are.

proj_𝐛(𝐚) = (𝐚·𝐛/|𝐛|²)·𝐛 — the component of 𝐚 along 𝐛.

When 𝐚·𝐛 = 0, meaning θ = 90°.

Any dimension, as long as both vectors match.

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CalculateMe Team

Last updated: 2026-07-15

Dot Product Calculator

Formula

How It Works

Examples

Dot product (1,2,3)·(4,5,6)

Projection of (3,4) onto (1,0)

Frequently Asked Questions

What is the dot product?

What is vector projection?

When are vectors orthogonal?

What dimensions are supported?

Related Calculators

Dot Product Calculator

Formula

How It Works

Examples

Dot product (1,2,3)·(4,5,6)

Projection of (3,4) onto (1,0)

Frequently Asked Questions

What is the dot product?

What is vector projection?

When are vectors orthogonal?

What dimensions are supported?

Related Calculators