Cross Product Calculator – Vector Cross Product
Calculate the cross product of two 3D vectors
Table of Contents
How to Use
- Enter the x, y, and z components of the first vector
- Enter the x, y, and z components of the second vector
- Click calculate to see the cross product result
- View the resulting vector and its magnitude
What is the Cross Product?
The cross product (also called vector product) is a binary operation on two vectors in three-dimensional space. It produces a vector that is perpendicular to both input vectors, with a magnitude equal to the area of the parallelogram formed by the two vectors.
Formula
For vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃):
A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
Properties of Cross Product
- Anti-commutative: A × B = -(B × A)
- Distributive: A × (B + C) = (A × B) + (A × C)
- Not associative: A × (B × C) ≠ (A × B) × C
- Perpendicular result: The result is perpendicular to both input vectors
- Zero for parallel vectors: If A and B are parallel, A × B = 0
- Magnitude: |A × B| = |A| |B| sin(θ), where θ is the angle between vectors
Real-World Applications
Physics
- Calculating torque (τ = r × F)
- Finding magnetic force (F = q(v × B))
- Angular momentum (L = r × p)
- Determining rotation axes
Engineering & Computer Graphics
- 3D modeling and rendering
- Surface normal calculations
- Collision detection
- Robotics and motion planning
- Game development
Mathematics
- Finding perpendicular vectors
- Calculating areas of parallelograms
- Determining plane equations
- Vector space operations
How to Calculate Cross Product
To calculate A × B where A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃):
- X component: a₂b₃ - a₃b₂
- Y component: a₃b₁ - a₁b₃
- Z component: a₁b₂ - a₂b₁
Example: A = (1, 2, 3) and B = (4, 5, 6)
- X = (2)(6) - (3)(5) = 12 - 15 = -3
- Y = (3)(4) - (1)(6) = 12 - 6 = 6
- Z = (1)(5) - (2)(4) = 5 - 8 = -3
Result: A × B = (-3, 6, -3)
Frequently Asked Questions
- What is the difference between dot product and cross product?
- The dot product produces a scalar (single number) and measures how much two vectors point in the same direction. The cross product produces a vector perpendicular to both input vectors and measures the area of the parallelogram they form.
- Why is the cross product only defined in 3D?
- The cross product is specifically defined for 3D vectors because it relies on the unique properties of three-dimensional space. While there are generalizations to other dimensions, the standard cross product operation is inherently three-dimensional.
- What does it mean if the cross product is zero?
- If A × B = 0, it means the vectors are parallel (pointing in the same or opposite directions). The magnitude of the cross product is |A| |B| sin(θ), which equals zero when θ = 0° or 180°.
- How do I determine the direction of the cross product?
- Use the right-hand rule: point your fingers in the direction of the first vector, curl them toward the second vector, and your thumb points in the direction of the cross product. The result is perpendicular to both input vectors.