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. It has applications 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 in x, y, z directions

3. Importance of Cross Product

Details: The cross product is essential for calculating torque, angular momentum, surface normals in 3D graphics, and determining if two vectors are parallel.

4. Using the Calculator

Tips: Enter the components of both vectors. The calculator supports both numeric values and variables (like x, y, a, b). The result will show the symbolic cross product formula.

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

Q4: What if my vectors have variables?
A: This calculator handles variables symbolically, showing the algebraic expression of the cross product.

Q5: What's 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.