1. What is 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 that the vectors span.

2. How Does the Calculator Work?

The calculator uses the determinant formula for cross product:

Which expands to:

• \( x \)-component: \( a_yb_z - a_zb_y \)
• \( y \)-component: \( a_zb_x - a_xb_z \)
• \( z \)-component: \( a_xb_y - a_yb_x \)

3. Applications of Cross Product

Details: Cross products are used in physics (torque, angular momentum), computer graphics (surface normals), engineering (moment of force), and mathematics (area calculations).

4. Using the Calculator

Tips: Enter the x, y, and z components for both vectors. The calculator will compute the resulting perpendicular vector.

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: Can you compute cross product in 2D?
A: Technically no, but you can extend 2D vectors to 3D by adding z=0, resulting in a vector with only z-component.

Q3: What does a zero cross product mean?
A: A zero cross product indicates that the vectors are parallel (or at least one is zero).

Q4: How is cross product related to area?
A: The magnitude of the cross product equals the area of the parallelogram formed by the two vectors.

Q5: What's the right-hand rule?
A: It's a mnemonic for determining the direction of the cross product vector - point fingers in direction of first vector, curl towards second vector, thumb points in cross product direction.