Cross Product Calculator
Vector A
Vector B
Understanding Cross Product
What is a Cross Product?
The cross product of two vectors results in a vector that is perpendicular to both input vectors. Its magnitude equals the area of the parallelogram formed by the two vectors.
Definition:
A × B = |A| |B| sin(θ) n
where:
- |A| and |B| are vector magnitudes
- θ is the angle between vectors
- n is the unit vector perpendicular to both A and B
Component Form:
A × B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k
Properties of Cross Product
Key Properties:
- Anti-commutative: A × B = -(B × A)
- Distributive: A × (B + C) = (A × B) + (A × C)
- Not associative: (A × B) × C ≠ A × (B × C)
- Scalar multiplication: (kA) × B = k(A × B)
Special Cases:
- Parallel vectors: A × B = 0
- Perpendicular vectors: |A × B| = |A| |B|
- Unit vectors: i × j = k, j × k = i, k × i = j
Geometric Applications
Area Calculations
Area of parallelogram = |A × B|
Area of triangle = ½|A × B|
Normal Vector
The cross product provides a vector perpendicular to a plane
Unit normal = (A × B)/|A × B|
Torque in Physics
τ = r × F
where r is position vector and F is force vector
Angular Momentum
L = r × p
where p is linear momentum
Applications in Different Fields
Computer Graphics
- Surface normal computation
- 3D model rendering
- Camera orientation
Robotics
- Robot arm orientation
- Motion planning
- Collision detection
Engineering
- Structural analysis
- Force calculations
- Moment of force