BravoCalc

Cross Product Calculator

Cross Product Calculator

Calculate and visualize the cross product of two 3D vectors

Vector A

Vector B

Result

[0.0000, 0.0000, 1.0000]
1.0000
90.00°

Formula Used

A × B = [AyBz - AzBy, AzBx - AxBz, AxBy - AyBx]
Vector A = [1, 0, 0]
Vector B = [0, 1, 0]
A × B = [0.00, 0.00, 1.00]

What is the Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both of the vectors being multiplied and normal to the plane containing them. The cross product is denoted by the symbol ×.

Cross Product Formula

For two vectors a = [a₁, a₂, a₃] and b = [b₁, b₂, b₃], the cross product a × b is defined as:

a × b = [a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁]

This can also be expressed using the determinant of a matrix:

a × b = |i j k |
         |a₁ a₂ a₃|
         |b₁ b₂ b₃|

Properties of the Cross Product

  • Anticommutativity: a × b = -(b × a)
  • Distributivity over addition: a × (b + c) = a × b + a × c
  • Scalar multiplication: (λa) × b = λ(a × b) = a × (λb)
  • Perpendicularity: a × b is perpendicular to both a and b
  • Magnitude: |a × b| = |a||b|sin(θ), where θ is the angle between a and b

Applications of the Cross Product

Physics

The cross product is used to calculate torque, angular momentum, and the magnetic force on a moving charge.

Computer Graphics

Cross products are essential for calculating surface normals, which are used in lighting calculations and determining the orientation of surfaces.

Engineering

Engineers use cross products in structural analysis, fluid dynamics, and electromagnetic calculations.

Robotics

Cross products help determine the axis of rotation for robotic arms and calculate the torque needed for movement.

How to Use the Cross Product Calculator

  1. Enter the x, y, and z components of the first vector (Vector A)
  2. Enter the x, y, and z components of the second vector (Vector B)
  3. Click the "Calculate Cross Product" button
  4. View the resulting cross product vector and its magnitude
  5. Explore the 3D visualization to better understand the spatial relationship

Did You Know?

The cross product only exists in 3 and 7 dimensions. In other dimensions, there is no binary operation on vectors that produces a vector perpendicular to the inputs with the appropriate magnitude.