Cross Product Calculator

Real-time 3D vector calculation tool with visualization

Cross Product Calculator

Vector A (First Vector)
i
j
k
Vector B (Second Vector)
i
j
k

Cross Product Result: A × B

Result Vector
-15, -2, 39
Magnitude (Length)
42.0
Unit Vector (Normalized)
-0.357, -0.048, 0.929
Calculation Steps
A × B = (a₂b₃ - a₃b₂)i + (a₃b₁ - a₁b₃)j + (a₁b₂ - a₂b₁)k
= ((-3×2) - (1×9))i + ((1×4) - (3×2))j + ((3×9) - (-3×4))k
= (-6 - 9)i + (4 - 6)j + (27 + 12)k
= -15i - 2j + 39k
Vector Visualization
Vector A
Vector B
A × B

Advanced Features

Real-Time Calculation

Results update instantly as you type. No need to press calculate button.

3D Visualization

Visual representation of vectors and their cross product in 3D space.

Step-by-Step Solution

Detailed calculation process showing each step of the cross product formula.

Magnitude & Direction

Calculate vector magnitude and normalized unit vector with direction.

Vector Swapping

Swap vectors A and B to see how cross product changes with order.

Export Results

Save or copy calculation results for reports and documentation.

Real-Time Updates 3D Vector Visualization Step-by-Step Solution Magnitude Calculation Unit Vector Vector Swapping Random Values One-Click Reset Formula Explanation Result Export Mobile Responsive Professional UI Error Handling SEO Optimized No Ads

Understanding Cross Product: A Comprehensive Guide

What is a Cross Product?

The cross product (also called vector product) is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both of the original vectors, making it extremely useful in physics, engineering, and computer graphics.

How to Use This Cross Product Calculator

  1. Enter Vector Components: Input the i, j, and k components for both Vector A and Vector B in the fields provided.
  2. Real-Time Calculation: As you type, the cross product will be calculated automatically and displayed in the results section.
  3. Visualize: Observe the visual representation of your vectors and their cross product in the 3D visualization box.
  4. Analyze Results: Review the resulting vector, its magnitude, unit vector, and the step-by-step calculation.
  5. Use Advanced Features: Swap vectors, generate random values, or save your results as needed.

The Cross Product Formula

For two vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), the cross product A × B is calculated as:

A × B = (a₂b₃ - a₃b₂)i + (a₃b₁ - a₁b₃)j + (a₁b₂ - a₂b₁)k

Practical Applications

  • Physics: Calculating torque, angular momentum, and magnetic force.
  • Engineering: Determining normal vectors to surfaces for structural analysis.
  • Computer Graphics: Creating surface normals for lighting calculations and 3D rendering.
  • Mathematics: Determining if vectors are parallel and finding areas of parallelograms.

Important Properties

  • The cross product is anti-commutative: A × B = - (B × A)
  • The result is perpendicular to both original vectors
  • The magnitude of the cross product equals the area of the parallelogram formed by the two vectors
  • If two vectors are parallel, their cross product is the zero vector
Pro Tip

Use the "Swap Vectors" button to verify the anti-commutative property of cross products. Notice how the result vector changes direction when you swap A and B.

Quick Keys
  • Tab - Navigate between fields
  • Enter - Calculate cross product
  • Esc - Reset all fields
  • R - Generate random vectors
  • S - Swap vectors A and B
Common Examples
  • Unit vectors: i × j = k
  • Parallel vectors: (1,2,3) × (2,4,6) = (0,0,0)
  • Perpendicular: (1,0,0) × (0,1,0) = (0,0,1)
  • 3D example: (2,3,4) × (5,6,7) = (-3,6,-3)
SEO Keywords
cross product vector calculator 3D vectors physics tool engineering math vector multiplication real-time calculator