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 important applications in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the cross product formula:

Where:

• \( \vec{u} = (u_x, u_y, u_z) \) — First input vector
• \( \vec{v} = (v_x, v_y, v_z) \) — Second input vector
• \( \vec{w} = (w_x, w_y, w_z) \) — Resulting cross product vector

Explanation: The cross product produces a vector that is perpendicular to both input vectors, with magnitude equal to the area of the parallelogram they span.

3. Importance of Cross Product

Details: The cross product is essential for calculating torque, angular momentum, surface normals in 3D graphics, and determining perpendicular directions in vector analysis.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of cross product?
A: The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, and its direction is perpendicular to both vectors.

Q2: Is the cross product commutative?
A: No, the cross product is anti-commutative: \( \vec{u} \times \vec{v} = -(\vec{v} \times \vec{u}) \).

Q3: What happens when vectors are parallel?
A: If two vectors are parallel, their cross product is the zero vector (0, 0, 0).

Q4: Can cross product be calculated in 2D?
A: In 2D, the cross product is a scalar representing the signed area of the parallelogram, but this calculator focuses on 3D vectors.

Q5: What are some practical applications?
A: Cross products are used in physics for torque calculations, in computer graphics for surface normals, and in engineering for moment calculations.