Cross Product Calculator
Calculate the cross product of two 3D vectors, producing a vector perpendicular to both inputs.
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.
How to Use This Calculator
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.
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.
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.
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.
Formula Reference
Cross Product Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
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.
About This Calculator
The Cross Product Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the cross product of two 3D vectors, producing a vector perpendicular to both inputs. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
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
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)
(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
Used in These Calculators
Calculators that build on or apply the concepts from this page:
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.