A × B = (y₁z₂ − z₁y₂, z₁x₂ − x₁z₂, x₁y₂ − y₁x₂)
Determinant Form:
| i j k |
| x₁ y₁ z₁ |
| x₂ y₂ z₂ |
Anti-commutative: A × B = −(B × A)
Perpendicular: Result is perpendicular to both input vectors
Magnitude: |A × B| = |A||B|sin(θ)
Area: |A × B| equals parallelogram area spanned by A and B
The cross product (also called the vector product) is a binary operation on two vectors in three-dimensional space. Unlike the dot product which produces a scalar, the cross product produces another vector that is perpendicular to both input vectors. This makes it extremely useful in physics and engineering for finding normal vectors, calculating torque, and determining angular momentum.
The direction of the resulting vector follows the right-hand rule: if you curl your fingers from vector A toward vector B, your thumb points in the direction of A × B. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, which can be calculated as |A||B|sin(θ) where θ is the angle between them.
Cross product calculations follow standard 3D vector formulas. Results depend on correct vector input. This calculator is for educational purposes and should be verified for critical applications in physics, engineering, or computer graphics.