1. What is Vector 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 they span.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

Where:

• \( \vec{A} = (A_x, A_y, A_z) \) — First vector
• \( \vec{B} = (B_x, B_y, B_z) \) — Second vector
• \( \hat{i}, \hat{j}, \hat{k} \) — Unit vectors in x, y, z directions

Explanation: The cross product is calculated using the determinant of a matrix composed of the unit vectors and the components of vectors A and B.

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, z components of both vectors. The calculator will compute the cross product which 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 result, while cross product gives a vector result perpendicular to both input vectors.

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 for cross product?
A: Point fingers in direction of first vector, curl towards second vector; thumb points in direction of cross product.

Q4: Can cross product be calculated in 2D?
A: In 2D, the cross product is treated as a scalar (z-component of the 3D cross product with z=0).

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