Cross Product Calculator for 3D Vectors

Calculate A × B from two 3D vectors. See the determinant expansion, magnitude, unit normal, angle, parallelogram and triangle areas, and perpendicularity checks. Everything runs locally in your browser.

Enter Two 3D Vectors

Vector A

Vector B

Accepts integers, decimals, scientific notation, fractions such as 1/2, and roots such as sqrt(3).

Cross Product A × B

—Enter six components and calculate.
Magnitude |A × B|—
Angle between A and B—
Parallelogram area—
Triangle area—
Unit normal in the A × B direction—
Perpendicularity check—

Advertisement

Step-by-Step Cross Product

The middle component includes its determinant sign automatically through AzBx − AxBz.

Calculate two valid vectors to see every substituted component and verification.

3D Cross Product Formula

For A = (Aₓ,Aᵧ,A_z) and B = (Bₓ,Bᵧ,B_z):

A × B = (AᵧB_z − A_zBᵧ, A_zBₓ − AₓB_z, AₓBᵧ − AᵧBₓ)

Magnitude and area

|A × B| = |A||B| sin(θ)

The magnitude is the parallelogram area. Divide by 2 for the triangle area.

Unit normal

n̂ = (A × B) / |A × B|

A unit normal exists only when the cross product is nonzero.

Perpendicularity

(A × B) · A = 0
(A × B) · B = 0

These dot products verify that the result is perpendicular to both inputs.

Numerical convention: Components are treated as coordinates or quantities sharing a compatible unit. Cross-product components and geometric areas therefore have squared units. Calculations use browser double-precision arithmetic, accept finite components no greater than 10¹⁰⁰ in magnitude, and do not round intermediate results.

How to Use the Cross Product Calculator

  1. Enter the x, y, and z components of vectors A and B.
  2. Optionally choose display precision and add a short unit label.
  3. Select Calculate cross product to evaluate A × B.
  4. Review the determinant steps, direction, angle, areas, unit normal, and dot-product checks. Swap the vectors to calculate B × A.

Worked Cross Product Example

For A = (1,2,3) and B = (4,5,6):

A × B = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3,6,−3)

Magnitude

√((−3)² + 6² + (−3)²) = √54 ≈ 7.3485, so the parallelogram area is about 7.3485 square units.

Direction matters

Swapping the order gives B × A = (3,−6,3). This is the same length but points in the opposite direction.

What a Cross Product Tells You

The right-hand rule determines the direction of A × B: curl the fingers of your right hand from A toward B through the smaller angle, and your thumb points along the result. Reversing the order reverses that direction.

A nonzero cross product gives a normal to the plane containing A and B. This makes it useful for surface normals, torque, angular momentum, orientation tests, and geometric areas. If the result is zero, the inputs do not determine a unique plane normal.

Cross Product Calculator FAQs

What is the cross product of two 3D vectors?

It is a vector perpendicular to both inputs. Its direction follows the right-hand rule, and its magnitude is the area of the parallelogram they span.

Does the order of a cross product matter?

Yes. The cross product is anti-commutative: B × A = −(A × B). Magnitudes and areas do not change when the order is reversed.

When is the cross product zero?

For two nonzero vectors, it is zero when they are parallel or antiparallel. It is also zero if either input is the zero vector.

How does the cross product give an area?

|A × B| = |A||B|sin(θ), which is base times perpendicular height for the parallelogram. Half of that is the triangle area.

Can I calculate a cross product in 2D?

Embed each 2D vector as (x,y,0). The x and y components of the result will be zero, while the z component is AₓBᵧ − AᵧBₓ.

Why is the unit normal sometimes undefined?

A zero cross product has no direction, so it cannot be divided by its magnitude to create a unit vector. This occurs for parallel vectors and whenever an input is zero.

Are my vector components private?

Yes. The calculator parses inputs, calculates, copies, and creates downloads locally. Its calculator code does not upload or store your values.

Calculation Notes

Last reviewed: August 5, 2026 by the Starlight Tools editorial team.

The calculator uses atan2(|A × B|, A · B) for a stable angle from 0° through 180°. Display precision affects shown values only. Near-parallel decimal vectors can produce a tiny nonzero cross product because of floating-point representation.

Explore more tools