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
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