Cross Product Calculator — 3D Vector Cross Product
Compute the cross product of two 3D vectors. Shows the resulting vector, magnitude, unit vector, and area of the parallelogram.
Cross Product
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The Formula
The cross product (also called the vector product) is a binary operation on two vectors in three-dimensional space. It produces a third vector that is perpendicular to both input vectors. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors: |a × b| = |a||b|sin θ. The direction follows the right-hand rule.
Variable Definitions
First Vector
The first 3D vector in the cross product operation.
Second Vector
The second 3D vector in the cross product operation.
Cross Product Vector
A vector perpendicular to both a and b, following the right-hand rule.
Magnitude
The length of the cross product, equal to |a||b|sin θ, also the area of the parallelogram.
Unit Vector
The normalized direction of the cross product, a × b / |a × b|.
How to Use This Calculator
- 1
Enter all three components of both 3D vectors a and b. All components are required.
- 2
The calculator computes the cross product a × b and displays the resulting vector components.
- 3
Review the magnitude, unit vector (normalized direction), and verification that the result is orthogonal to both inputs.
- 4
The area result shows the area of the parallelogram spanned by the two vectors, which equals the cross product magnitude.
Quick Reference
| From | To |
|---|---|
| i × j = k | Standard basis vectors follow a cyclic pattern |
| j × k = i | Right-hand rule determines direction |
| k × i = j | Reversing order flips the sign |
| Perpendicular check | (a × b) · a = 0 and (a × b) · b = 0 |
Common Applications
- Physics: calculating torque (τ = r × F) where r is the position vector and F is the force vector.
- Electromagnetism: computing the Lorentz force (F = qv × B) on a charged particle in a magnetic field.
- Computer graphics: calculating surface normals for lighting calculations by taking cross products of edge vectors.
- Robotics: computing angular momentum (L = r × p) and rotational dynamics.
- Engineering: determining the moment of a force about a point, essential for structural analysis and mechanical design.
The cross product of two vectors produces a third vector perpendicular to both, with magnitude equal to the parallelogram area
Understanding the Concept
The cross product is a fundamental vector operation in three-dimensional space. Unlike the dot product which produces a scalar, the cross product produces a vector. The resulting vector is always perpendicular (orthogonal) to both input vectors, making it an indispensable tool for constructing normal vectors. The magnitude |a × b| = |a||b|sin θ gives the area of the parallelogram formed by the two vectors, and equals zero when the vectors are parallel (θ = 0° or 180°). The direction follows the right-hand rule: if you curl the fingers of your right hand from a toward b, your thumb points in the direction of a × b. The cross product is anti-commutative: a × b = −(b × a), meaning swapping the inputs reverses the direction. The cross product is intimately connected to the determinant of a 3×3 matrix formed by the standard basis vectors and the components of a and b.
Frequently Asked Questions
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