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. Its magnitude relates to the area of the parallelogram spanned by the two vectors.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

Features:

• Handles both constant vectors and time-dependent vector functions
• Accepts numerical values or algebraic expressions (like '2t' or 'sin(t)')
• Shows the complete cross product calculation

3. Applications of Cross Product

Physics: Torque, angular momentum, magnetic force (Lorentz force).
Engineering: Moment of force, rotation calculations.
Computer Graphics: Surface normal calculations, shading.

4. Using the Calculator

Steps:

1. Enter components for both vectors A and B
2. Optionally enter a time value for time-dependent vectors
3. Click "Calculate" to see the cross product result

Note: For time-dependent vectors, you can enter expressions like "2t" or "sin(t)".

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar result, while cross product gives a vector perpendicular to both input vectors.

Q2: Can I calculate cross product in 2D?
A: The cross product is inherently a 3D operation. For 2D vectors, you can add a z-component of 0.

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

Q4: How is the direction of the cross product determined?
A: By the right-hand rule - point fingers in direction of A, curl towards B, thumb points in A×B direction.

Q5: Can I use this for vector functions?
A: Yes, enter components as functions of t (like "2t" or "cos(t)") and specify a time value if needed.