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 spanned by them.

2. How Does the Calculator Work?

The calculator uses the determinant formula:

Which expands to:

• X-component: \( A_yB_z - A_zB_y \)
• Y-component: \( A_zB_x - A_xB_z \)
• Z-component: \( A_xB_y - A_yB_x \)

Explanation: The cross product produces a vector perpendicular to both input vectors, with magnitude equal to the product of their magnitudes and the sine of the angle between them.

3. Importance of Cross Product

Details: The cross product is essential in physics (torque, angular momentum), engineering (moment of force), and computer graphics (surface normals).

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The result will be a new vector in 3D space 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 result, while cross product gives a vector result perpendicular to both inputs.

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

Q3: Can you do cross product in 2D?
A: The cross product is only defined in 3D and 7D spaces. In 2D, you can compute a similar scalar value.

Q4: What's the right-hand rule?
A: A mnemonic for determining the direction of the cross product vector - point fingers in direction of A, curl toward B, thumb points in result direction.

Q5: When is the cross product zero?
A: When vectors are parallel or one is zero, as sin(0°) = 0.