1. What is Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space that results in a vector perpendicular to both original vectors. It's widely used in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

Where:

• \( \mathbf{A} = (A_x, A_y, A_z) \) — First vector
• \( \mathbf{B} = (B_x, B_y, B_z) \) — Second vector
• \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) — Unit vectors

Explanation: The cross product magnitude equals the area of the parallelogram formed by the two vectors, and its direction follows the right-hand rule.

3. Importance of Cross Product

Details: Cross products are essential for calculating torque, angular momentum, surface normals in 3D graphics, and determining if vectors are parallel.

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1,2,3"). Both vectors must have exactly 3 components. The calculator will compute the perpendicular vector.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar result, while cross product gives a vector perpendicular to both input vectors.

Q2: Can I calculate cross product in 2D?
A: The standard cross product is defined for 3D space. For 2D vectors, you can treat them as 3D with z=0.

Q3: What does a zero cross product mean?
A: A zero cross product indicates the vectors are parallel (or at least one is zero).

Q4: How is cross product used in real applications?
A: Used in physics (torque), computer graphics (normal vectors), and engineering (moment calculations).

Q5: What's the right-hand rule?
A: Point fingers in direction of first vector, curl toward second vector - thumb points in cross product direction.