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
• The result is a new vector perpendicular to both a and b

Explanation: The cross product magnitude equals the area of the parallelogram formed by the two vectors, and its direction follows the right-hand rule.

3. Importance of Cross Product

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

4. Using the Calculator

Tips: Enter the x, y, z components for both vectors. The calculator will compute the resulting vector that is 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 quantity, while cross product gives a vector quantity perpendicular to both input vectors.

Q2: What does a zero cross product mean?
A: A zero cross product indicates that the vectors are parallel (or at least one is a zero vector).

Q3: Can I calculate cross product in 2D?
A: The standard cross product is defined for 3D space. For 2D vectors, you can treat them as 3D with z=0, resulting in a vector with only z-component.

Q4: What's the right-hand rule?
A: Point your index finger along vector a, middle finger along vector b, and your thumb will point in the direction of a × b.

Q5: How is cross product used in physics?
A: It's used to calculate torque (τ = r × F), angular momentum (L = r × p), and magnetic force (F = qv × B).