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 spanned by the two vectors.

2. How Does the Calculator Work?

The calculator uses the cross product formula:

Where:

• \( \vec{A} = (A_x, A_y, A_z) \) — First vector components
• \( \vec{B} = (B_x, B_y, B_z) \) — Second vector components
• × — Cross product operator

Explanation: The result is a new vector perpendicular to both A and B, following the right-hand rule.

3. Importance of Cross Product

Details: The cross product is essential in physics (torque, angular momentum), engineering (moment of force), and computer graphics (surface normals).

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The calculator will compute the resulting 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: \( |\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta \).

Q3: Can I calculate cross product in 2D?
A: For 2D vectors, treat them as 3D with z=0. The result will have only a z-component.

Q4: What's the right-hand rule?
A: Point fingers in direction of A, curl towards B; thumb points in direction of A×B.

Q5: What if the cross product is zero?
A: A zero vector result means the input vectors are parallel (or at least one is zero).