New: AI Infographics — turn data and ideas into shareable graphics from one prompt. Try it now
Cross Product Calculator
Calculate the cross product of vectors in any dimension instantly. Supports 2D, 3D, and higher-dimensional vectors with flexible input using commas or spaces.
The cross product is a fundamental operation in vector mathematics. This calculator supports vectors of any dimension, with specialized handling for 2D and 3D cases.
Input Format
Enter vector components separated by commas or spaces:
- Example with commas: 1, 2, 3
- Example with spaces: 1 2 3
- Example mixed: 1, 2 3, 4
Dimension-Specific Behavior
- 2D vectors: Returns the z-component (scalar) representing signed area
- 3D vectors: Returns standard 3D cross product vector
- Higher dimensions: Returns wedge product components
3D Cross Product Formula
A × B = (A_y·B_z - A_z·B_y, A_z·B_x - A_x·B_z, A_x·B_y - A_y·B_x)
Properties
- Anti-commutative: A × B = -(B × A)
- Distributive: A × (B + C) = A × B + A × C
- The magnitude |A × B| equals the area of the parallelogram with sides A and B
- If A and B are parallel, their cross product is zero
Common Applications
- Computing surface normals in 3D graphics
- Calculating torque in physics
- Finding the area of triangles and parallelograms
- Determining the orientation of objects in space
- Solving systems of linear equations
Understanding Cross Products
The cross product is a binary operation on two vectors in three-dimensional space that results in a vector perpendicular to both input vectors. This mathematical operation is essential in physics, engineering, computer graphics, and many other fields.
How to Use This Calculator
- Enter the X, Y, and Z components of your first vector (Vector A)
- Enter the X, Y, and Z components of your second vector (Vector B)
- Click "Compute Cross Product" to see the result
- The calculator will display both the resulting vector and its magnitude
Applications of Cross Products
- Physics: Calculating torque, angular momentum, and magnetic force
- Computer Graphics: Computing surface normals for lighting calculations
- Engineering: Analyzing forces and moments in mechanical systems
- Mathematics: Finding areas of parallelograms and solving geometric problems
Mathematical Properties
The cross product has several important properties that make it useful in various applications:
- The result is always perpendicular to both input vectors
- The magnitude equals the area of the parallelogram formed by the two vectors
- The direction follows the right-hand rule
- If vectors are parallel, the cross product is zero
Found This Tool Helpful?
Share it with others in one click.