Vector Cross Product Calculator

Category: Linear Algebra
- May 20, 2025
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Calculate the cross product of two 3D vectors. The cross product results in a vector that is perpendicular to both input vectors, with a magnitude equal to the area of the parallelogram they form.

First Vector (A)

Second Vector (B)

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Advanced Options

Understanding Vector Cross Products

The cross product of two vectors A and B, denoted as A ร— B, is a vector that is perpendicular to both A and B. The cross product is used extensively in physics, engineering, computer graphics, and many other fields.

Cross Product Formula

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

A ร— B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

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

A ร— B = | i j k a1 a2 a3 b1 b2 b3 |
Key Properties
  • Perpendicularity: A ร— B is perpendicular to both A and B
  • Magnitude: |A ร— B| = |A||B|sin(ฮธ) where ฮธ is the angle between A and B
  • Anti-commutativity: A ร— B = -(B ร— A)
  • Distributivity: A ร— (B + C) = A ร— B + A ร— C
  • Not associative: A ร— (B ร— C) โ‰  (A ร— B) ร— C
  • Scalar multiplication: (kA) ร— B = k(A ร— B) = A ร— (kB)
Applications of Cross Products
  • Physics: Calculating torque, angular momentum, and magnetic force
  • Computer Graphics: Determining surface normals and camera orientations
  • Geometry: Finding perpendicular vectors and calculating areas
  • Navigation: Computing orientations in 3D space
  • Robotics: Joint movement calculations and path planning

Disclaimer: This calculator performs vector cross product operations for educational and computational purposes. The visualization is an approximate representation and may not be to exact scale for all vectors.

What Is the Vector Cross Product Calculator?

The Vector Cross Product Calculator is a simple and efficient tool for computing the cross product of two vectors in 3D space. This operation is useful in many areas such as Physics, engineering, robotics, and computer graphics. The result is a third vector that is perpendicular to both input vectors and whose magnitude represents the area of the parallelogram they form.

Cross Product Formula

A ร— B = (aโ‚‚bโ‚ƒ โˆ’ aโ‚ƒbโ‚‚, aโ‚ƒbโ‚ โˆ’ aโ‚bโ‚ƒ, aโ‚bโ‚‚ โˆ’ aโ‚‚bโ‚)

This calculation helps you find a vector that is orthogonal to both A and B, which is vital in 3D space applications.

How to Use the Calculator

Follow these steps to get accurate results:

  • Enter the X, Y, and Z components for Vector A and Vector B.
  • Select your preferred vector notation: component form, unit vector form, or column vector form.
  • Choose the number of decimal places to round your results.
  • Optional: Enable normalization to convert the result into a unit vector.
  • Click the "Calculate Cross Product" button to view the result.
  • Use the "Reset" button to clear inputs and start over.

Why Use This Calculator?

This tool is more than just a cross product solverโ€”it provides detailed insights and visualizations to deepen your understanding of vector relationships.

  • Visualize vectors and the resulting cross product on a chart.
  • Understand Geometry with calculated angles and magnitudes.
  • Explore properties such as perpendicularity, parallelism, and vector area.
  • Learn applications in torque, angular momentum, and surface normals.

Frequently Asked Questions (FAQ)

What is a cross product in vector Math?

The cross product of two 3D vectors produces another vector that is perpendicular to both. It's useful in calculating area, torque, and orientation in space.

Can this calculator show the steps?

Yes. When the "Show calculation details" option is selected, the calculator explains each step and shows how the cross product is derived.

What does "normalize result" mean?

Normalization scales the resulting vector to have a length (magnitude) of 1. This is especially useful for directional analysis and unit vector representation.

Is this calculator useful for Other vector operations?

Yes. While focused on cross products, this calculator complements other tools like the Vector Addition Calculator, Unit Vector Calculator, and Vector Projection Calculator for a complete vector analysis experience.

How is this different from matrix calculators?

Unlike matrix-specific tools such as the LU Decomposition Calculator or the Diagonalize Matrix Calculator, this calculator is specialized for 3D vector operations. It's ideal when your focus is spatial calculations rather than solving systems or transforming matrices.

How This Calculator Helps You

This tool simplifies the process of computing cross products and understanding their implications. Whether youโ€™re solving physics problems, developing simulations, or learning Linear Algebra concepts, it provides both the answer and the explanation. Combined with other utilities like the Dot Product Calculator or Triple Scalar Product Calculator, it forms part of a comprehensive toolkit for vector and matrix operations.