How to Calculate Cross Product
What is Cross Product?
The cross product of two 3D vectors produces a third vector perpendicular to both. Its magnitude equals the area of the parallelogram formed by the two vectors. It is used in physics, 3D graphics, and engineering.
Formula
u × v = |u||v|sin(θ)n̂; in 3D: (uᵧvᵧ − uᵧvᵧ, uᵧvₓ − uₓvᵧ, uₓvᵧ − uᵧvₓ)
- u, v
- two 3D vectors
- |u|, |v|
- magnitudes of vectors
- θ
- angle between vectors
- n̂
- unit normal vector perpendicular to both
Step-by-Step Guide
- 1A × B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx)
- 2|A × B| = |A||B|sin(θ)
- 3Result is perpendicular to both A and B
- 4Right-hand rule determines direction
Worked Examples
Input
A=(1,0,0), B=(0,1,0)
Result
A×B = (0,0,1) — unit z vector
Input
A=(1,2,3), B=(4,5,6)
Result
(−3,6,−3)
Frequently Asked Questions
What is the right-hand rule for cross product?
Point fingers along u, curl toward v, thumb points in direction of u × v.
Is the cross product commutative?
No! u × v = −(v × u). Order matters, and swapping reverses the direction.
What is the geometric meaning of cross product magnitude?
|u × v| equals the area of the parallelogram formed by u and v.
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