Vector Projection Calculator

Use our free vector projection calculator to compute proj_𝐛(𝐚), the scalar projection comp_𝐛(𝐚), and the perpendicular component 𝐚_⟂ with full step-by-step verification.

Math

Mode

𝐚 =

𝐛 =

Formula

Decompose 𝐚 into parallel and perpendicular components relative to 𝐛. Compute the vector projection, scalar projection (component), and perpendicular component with verification of orthogonality and Pythagoras.

proj_𝐛(𝐚) = (𝐚·𝐛/|𝐛|²)·𝐛, comp_𝐛(𝐚) = 𝐚·𝐛/|𝐛|

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: vector projection, scalar projection, or perpendicular component.
3Click Calculate.

Examples

Project (3,4,0) onto (1,0,0)

1.proj = ((3·1+4·0+0·0)/(1²+0²+0²))·(1,0,0)
2.= 3·(1,0,0) = (3,0,0)

Result: 3

Scalar projection of (3,4) onto (1,1)

1.comp = (3·1+4·1)/√2 = 7/√2 ≈ 4.95

Result: 4.9497474683

Frequently Asked Questions

The component of 𝐚 that lies along 𝐛: proj_𝐛(𝐚) = (𝐚·𝐛/|𝐛|²)·𝐛.

The signed length of the projection: comp_𝐛(𝐚) = 𝐚·𝐛/|𝐛|.

𝐚_⟂ = 𝐚 − proj_𝐛(𝐚), which is orthogonal to 𝐛.

|𝐚|² = |proj|² + |𝐚_⟂|² — confirming orthogonal decomposition.

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CalculateMe Team

Last updated: 2026-07-15

Vector Projection Calculator

Formula

How It Works

Examples

Project (3,4,0) onto (1,0,0)

Scalar projection of (3,4) onto (1,1)

Frequently Asked Questions

What is vector projection?

What is scalar projection?

What is the perpendicular component?

What is the Pythagoras check?

Related Calculators

Vector Projection Calculator

Formula

How It Works

Examples

Project (3,4,0) onto (1,0,0)

Scalar projection of (3,4) onto (1,1)

Frequently Asked Questions

What is vector projection?

What is scalar projection?

What is the perpendicular component?

What is the Pythagoras check?

Related Calculators