Cross Product Calculator Online— Quick Guide & How to Use

The cross product is a core operation in vector algebra that produces a vector perpendicular to two input vectors. Whether you’re a student, engineer, or developer, a fast cross product calculator makes solving 2D and 3D problems quick and error-free.

What is the Cross Product?

For two vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz), the cross product A × B is:
$A × B = (Ay·Bz − Az·By,\n Az·Bx − Ax·Bz,\n Ax·By − Ay·Bx)$

This operation returns a vector orthogonal to both A and B and its magnitude equals the area of the parallelogram spanned by A and B:
$|A × B| = |A||B|sin(θ)$

When to use a Cross Product Calculator

• Quickly computing orthogonal vectors for physics and 3D graphics.
• Solving torque, angular momentum, and normal vector problems.
• Verifying hand calculations or classroom exercises.

How to use the Cross Product Calculator (step-by-step)

1. Enter components for Vector A and Vector B. For 2D vectors, set Az = 0 and Bz = 0.
2. Click Calculate to get the result vector and its magnitude.
3. Use the Copy button to paste results into notes or reports.

Example

If A = (1, 2, 3) and B = (4, 5, 6) then:
$A × B = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (12 − 15, 12 − 6, 5 − 8) = (−3, 6, −3)$

Quick reference table

Input type	Notes	Result
2D vectors	Set z components to 0	Produces vector perpendicular to plane (z-only for 2D)
Parallel vectors	θ = 0 → sin θ = 0	Result = (0,0,0)
Perpendicular vectors	θ = 90° → sin θ = 1	Magnitude =

Many vector problems require simplifying algebraic expressions before finding the cross product. Try our Algebra Calculator for step-by-step equation solving.

Related tools and keywords

• vector cross product calculator: for general vector inputs.
• cross product of two vectors calculator / cross product calculator 2 vectors: typical student queries.
• matrix cross product calculator: uses determinant form to compute components.
• how to calculate cross product using calculator: explains step-by-step entry and validation.

Best practices

• Double-check units and signs.
• For symbolic or matrix workflows, use determinant notation to cross-check numeric results.
• Use the calculator when speed and accuracy are needed — especially in labs and programming tasks.

To learn more about vector cross products, vector algebra, and their applications in mathematics and physics, visit Wolfram MathWorld – Cross Product.