Cross Product Calculator

Q: Is it commutative?

No — 𝐚×𝐛 = −(𝐛×𝐚).

Use our free cross product calculator to compute 𝐚×𝐛 for 3D vectors, the area of the parallelogram spanned by 𝐚 and 𝐛, the triangle area, and the unit normal direction.

Math

Mode

𝐚 =

𝐛 =

Formula

Compute the 3D cross product 𝐚×𝐛 — the vector perpendicular to both 𝐚 and 𝐛. Also compute the parallelogram area, triangle area, and the unit normal direction using the right-hand rule.

𝐚×𝐛 = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁)

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Enter two 3D vectors (exactly 3 components each).
2Select mode: cross product, area, or direction.
3Click Calculate.

Examples

(1,2,3)×(4,5,6)

1.(2·6−3·5, 3·4−1·6, 1·5−2·4)
2.= (12−15, 12−6, 5−8) = (−3, 6, −3)

Result: 7.3484692283

Parallelogram area

1.|(1,0,0)×(0,1,0)| = |(0,0,1)| = 1

Result: 1

Frequently Asked Questions

A vector perpendicular to both inputs, only defined in 3D.

The area of the parallelogram spanned by 𝐚 and 𝐛.

Point fingers along 𝐚, curl toward 𝐛; thumb points in direction of 𝐚×𝐛.

𝐚×𝐛 = 𝟎 (zero vector) — the cross product of parallel vectors is zero.

No — 𝐚×𝐛 = −(𝐛×𝐚).

Related Calculators

Vector Calculator Vector Addition Calculator Vector Subtraction Calculator Angle Between Vectors Calculator Percentage Calculator Basic Calculator

CalculateMe Team

Last updated: 2026-07-15

Cross Product Calculator

Formula

How It Works

Examples

(1,2,3)×(4,5,6)

Parallelogram area

Frequently Asked Questions

What is the cross product?

What does |𝐚×𝐛| represent?

What is the right-hand rule?

What if vectors are parallel?

Is it commutative?

Related Calculators

Cross Product Calculator

Formula

How It Works

Examples

(1,2,3)×(4,5,6)

Parallelogram area

Frequently Asked Questions

What is the cross product?

What does |𝐚×𝐛| represent?

What is the right-hand rule?

What if vectors are parallel?

Is it commutative?

Related Calculators