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:

Where:

• \( \mathbf{A} = (A_x, A_y, A_z) \) — First vector
• \( \mathbf{B} = (B_x, B_y, B_z) \) — Second vector
• \( \times \) — Cross product operator

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. Applications of Cross Product

Details: The cross product is used in physics (torque, angular momentum), computer graphics (surface normals), engineering (moment of forces), and mathematics (determining orthogonality).

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 and measures projection, while cross product gives a vector quantity and measures perpendicularity.

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

Q3: Is the cross product commutative?
A: No, it's anti-commutative: A × B = - (B × A).

Q4: Can you compute cross product in 2D?
A: The standard cross product is defined for 3D, but a similar operation in 2D gives a scalar representing the signed area.

Q5: How is the magnitude related to the angle between vectors?
A: ||A × B|| = ||A|| ||B|| sin(θ), where θ is the angle between them.