1. What is the 3D 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 has applications in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

Which expands to: \[ (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k} \]

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: The cross product is essential for calculating torque, angular momentum, surface normals in 3D graphics, and determining perpendicular vectors in 3D space.

4. Using the Calculator

Tips: Enter the x, y, and z components for both vectors A and B. The calculator will compute the resulting perpendicular vector.

5. Frequently Asked Questions (FAQ)

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

Q2: What does the cross product magnitude represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q3: Why is the cross product only defined in 3D?
A: The perpendicular vector concept only works in 3D. In 2D, the cross product gives a scalar (the z-component of what would be a 3D result).

Q4: What's the right-hand rule?
A: Point fingers in direction of first vector, curl them toward second vector - thumb points in cross product direction.

Q5: Can the cross product be zero?
A: Yes, when vectors are parallel or one/both are zero vectors.