1. What is the 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. Its magnitude relates to the area of the parallelogram formed by the two vectors.

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
• The result is a new vector perpendicular to both A and B

3. Applications of Cross Product

Details: The cross product is used in physics (torque, angular momentum), computer graphics (surface normals), engineering (moment of forces), and mathematics (determining orthogonality).

4. Using the Calculator

Tips: Enter the x, y, and z components of both vectors. The calculator will compute the cross product vector components.

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 quantity perpendicular to both input vectors.

Q2: What does the magnitude of the cross product 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 uniquely works in three dimensions. In 2D, the result would be a scalar, and in higher dimensions, the concept generalizes differently.

Q4: What is the right-hand rule?
A: It's a mnemonic for determining the direction of the cross product vector - point fingers in direction of A, curl toward B, thumb points in direction of A × B.

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