1. What is the 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 formed by the two vectors.

2. How Does the Calculator Work?

The calculator uses the cross product formula:

Where:

• \( A = (A_x, A_y, A_z) \) — First vector
• \( B = (B_x, B_y, B_z) \) — Second vector
• \( A × B \) — Resulting perpendicular vector

Explanation: The cross product produces a vector that is orthogonal to both input vectors, with magnitude proportional to the sine of the angle between them.

3. Applications of Cross Product

Details: The cross product is 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 cross product 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. Dot product measures parallelness, cross product measures perpendicularity.

Q2: What does the magnitude of the cross product represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q3: What is the right-hand rule?
A: 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.

Q4: Can cross product be calculated in 2D?
A: In 2D, the cross product is a scalar representing the signed area of the parallelogram. It's essentially the z-component of what would be the 3D cross product.

Q5: What happens if vectors are parallel?
A: The cross product of parallel vectors is the zero vector (0, 0, 0).