Enter Vectors

Vector A

Vector B

Results

Cross Product (A × B)

Magnitude

Unit Vector

Angle Between Vectors

Parallelogram Area

The magnitude of the cross product equals the area of the parallelogram formed by vectors A and B.

Step-by-Step Solution

Vector Properties

3D Visualization

About Cross Product

What is a Cross Product?

The cross product (or vector product) of two vectors A and B is a vector perpendicular to both A and B. It's denoted as A × B and is defined only in three-dimensional space.

Formula

A × B = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k

Properties

  • Anticommutative: A × B = -(B × A)
  • Distributive: A × (B + C) = A × B + A × C
  • Perpendicular: The result is perpendicular to both input vectors
  • Magnitude: |A × B| = |A| |B| sin(θ)
  • Zero Vector: A × A = 0

Applications

  • Physics: Torque, angular momentum, magnetic force
  • Computer Graphics: Surface normals, lighting calculations
  • Engineering: Moment of force calculations
  • Geometry: Area of parallelograms and triangles