Cross Product Calculator
Calculate the cross product of two 3D vectors with magnitude and direction
Cross Product (A × B)
Unit Vector
Direction of result
|A| (Magnitude)
Length of Vector A
|B| (Magnitude)
Length of Vector B
Component Breakdown
Formula Used
A × B = (A₂B₃ - A₃B₂, A₃B₁ - A₁B₃, A₁B₂ - A₂B₁)
The result is perpendicular to both input vectors (right-hand rule)
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About Cross Product Calculator
What is the Cross Product?
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 a vector that is perpendicular to both input vectors. This makes it essential in physics and engineering for calculating torque, angular momentum, and surface normals.
Cross Product Formula
For two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the cross product is:
a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
This can also be computed using a 3×3 determinant with unit vectors i, j, k:
| i j k |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ |
Magnitude of the Cross Product
The magnitude of the cross product equals the area of the parallelogram formed by the two vectors:
|a × b| = |a| × |b| × sin(θ)
Where θ is the angle between the two vectors.
How to Use This Calculator
- Enter Vector A Components: Input the x, y, and z values for the first vector.
- Enter Vector B Components: Input the x, y, and z values for the second vector.
- View Results: The calculator instantly shows the cross product vector, its magnitude, and the unit vector.
Properties of the Cross Product
- Perpendicular Result: The result is always perpendicular to both input vectors (follows the right-hand rule).
- Non-commutative: a × b = -(b × a), meaning order matters.
- Distributive: a × (b + c) = a × b + a × c
- Scalar Multiplication: (ka) × b = k(a × b)
- Zero Result: Parallel vectors give a zero cross product.
Applications of the Cross Product
- Physics: Calculating torque (τ = r × F) and angular momentum.
- Engineering: Finding surface normals for 3D modeling.
- Computer Graphics: Determining face orientations for lighting.
- Navigation: Computing perpendicular directions.
Frequently Asked Questions
Why is the cross product only defined in 3D?
The cross product as traditionally defined only works in 3D because only in 3D does the concept of a unique perpendicular direction to two vectors exist.
What does a zero cross product mean?
A zero cross product indicates the two vectors are parallel (or one is the zero vector). There's no unique perpendicular direction.
How do I determine the direction of the result?
Use the right-hand rule: point your fingers in the direction of vector a, curl them toward vector b, and your thumb points in the direction of a × b.
Note: This calculator uses standard right-hand coordinate system conventions. The cross product is fundamental in vector calculus and has extensive applications in physics and engineering.