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's widely used in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the determinant 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 is calculated by expanding the determinant using the first row (i, j, k unit vectors).

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 vector components as comma-separated symbolic expressions (e.g., "a, b, c" or "2*x, 3*y, z^2"). The calculator will compute the symbolic cross product without simplifying.

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: Can I use this for 2D vectors?
A: For 2D vectors, treat them as 3D with z=0. The result will have only a z-component.

Q3: What does the magnitude of the cross product represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q4: Why is the cross product anti-commutative?
A: Because A × B = - (B × A), which follows from the determinant properties.

Q5: Can I calculate cross product for symbolic vectors?
A: Yes, this calculator is designed specifically for symbolic computation.