1. What is the Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both original vectors, with magnitude equal to the area of the parallelogram they span.

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 along x, y, z axes

Explanation: The cross product is calculated using the determinant of a 3×3 matrix with the unit vectors in the first row.

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 result will be a new vector perpendicular to both input vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar, while cross product gives a vector. Dot product measures parallelism, cross product measures perpendicularity.

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: Is cross product commutative?
A: No, A × B = - (B × A). It's anti-commutative.

Q4: What's the cross product of parallel vectors?
A: The zero vector (0, 0, 0), since sin(0°) = 0.

Q5: Can you compute cross product in 2D?
A: Not directly, but you can treat 2D vectors as having z=0 and get a result with only z-component non-zero.