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Cross Product Calculator

Calculate the cross product of two 3D vectors, producing a vector perpendicular to both inputs.

Reviewed by Christopher FloiedPublished Updated

This free online cross product calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

Result x

-3

Result y

6

Result z

-3

Magnitude (parallelogram area)

7.348469

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Cross Product Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Cross Product Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Cross Product Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Cross Product Calculator

The cross product takes two 3D vectors and produces a third vector that is perpendicular to both. The magnitude of the resulting vector equals the area of the parallelogram formed by the two input vectors. The direction follows the right-hand rule: point your right hand's fingers from A toward B and your thumb points in the direction of A x B. The cross product is essential in physics for computing torque, angular momentum, and magnetic force. In computer graphics, it is used to compute surface normals for lighting calculations. In engineering, it helps analyze forces and moments in three dimensions. Unlike the dot product, the cross product is not commutative: A x B = -(B x A).

The Math Behind It

The cross product A x B for 3D vectors A = (x1, y1, z1) and B = (x2, y2, z2) is defined as the vector (y1*z2 - z1*y2, z1*x2 - x1*z2, x1*y2 - y1*x2). This can be remembered using the determinant of a 3x3 matrix with i, j, k unit vectors in the first row and the components of A and B in the second and third rows. The magnitude |A x B| = |A| × |B| × sin(θ), where θ is the angle between the vectors. This gives the area of the parallelogram spanned by A and B. When the vectors are parallel, sin(θ) = 0 and the cross product is the zero vector. The cross product is anti-commutative (A x B = -B x A), distributive over addition, and not associative. It is only defined in 3D (and 7D, in advanced mathematics). The scalar triple product A . (B x C) gives the volume of the parallelepiped formed by three vectors. The cross product is fundamental to differential geometry (surface normals), electromagnetism (Lorentz force: F = qv x B), and fluid dynamics (vorticity).

Formula Reference

Cross Product

A × B = (y₁z₂ - z₁y₂, z₁x₂ - x₁z₂, x₁y₂ - y₁x₂)

Variables: A = (x₁, y₁, z₁), B = (x₂, y₂, z₂)

Worked Examples

Example 1: Cross product of two 3D vectors

Find (1, 2, 3) × (4, 5, 6)

Step 1:x: 2×6 - 3×5 = 12 - 15 = -3
Step 2:y: 3×4 - 1×6 = 12 - 6 = 6
Step 3:z: 1×5 - 2×4 = 5 - 8 = -3
Step 4:Result vector: (-3, 6, -3)
Step 5:Magnitude: √(9 + 36 + 9) = √54 ≈ 7.348

(1, 2, 3) × (4, 5, 6) = (-3, 6, -3), magnitude ≈ 7.348

Common Mistakes & Tips

  • !Getting the sign pattern wrong in the components (the y component has a reversed subtraction order).
  • !Confusing the cross product with the dot product.
  • !Forgetting the anti-commutative property (A × B = -(B × A)).
  • !Trying to compute the cross product in 2D (it is only defined in 3D).

Related Concepts

Dot Product

Produces a scalar measuring alignment, complementary to the cross product.

Vector Magnitude

The magnitude of the cross product gives the parallelogram area.

Matrix Determinant

The cross product formula uses a 3×3 determinant structure.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Dot Product Calculator

Unit Vector Calculator

Frequently Asked Questions

Is the cross product commutative?

No. A × B = -(B × A). This is called anti-commutativity.

What does the magnitude of the cross product represent?

The area of the parallelogram formed by the two input vectors.

Can I compute the cross product in 2D?

Not directly. The cross product is defined for 3D vectors. For 2D vectors, you can extend them to 3D by setting z = 0; the result will be a vector pointing in the z direction.

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