1. What is a 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. It has applications in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

Where:

• \( \vec{a} \) — First vector (a_x, a_y, a_z)
• \( \vec{b} \) — Second vector (b_x, b_y, b_z)
• \( \hat{i}, \hat{j}, \hat{k} \) — Unit vectors in x, y, z directions

Explanation: The cross product magnitude equals the area of the parallelogram formed by the two vectors, and its direction follows the right-hand rule.

3. Properties of Cross Product

Details: The cross product is anti-commutative (a × b = −b × a), distributive over addition, and compatible with scalar multiplication. It produces a vector perpendicular to both input vectors.

4. Using the Calculator

Tips: Enter the x, y, z components of both vectors. The calculator will compute the resulting vector components. All fields are required.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between cross product and dot product?
A: Cross product yields a vector perpendicular to both inputs, while dot product yields a scalar representing their parallel component.

Q2: When is the cross product zero?
A: When vectors are parallel or one/both are zero vectors (collinear).

Q3: Can you calculate cross product in 2D?
A: Technically no, but you can treat 2D vectors as having z=0, resulting in a vector with only z-component.

Q4: What's the right-hand rule?
A: Point fingers in direction of first vector, curl toward second vector; thumb points in cross product direction.

Q5: What are practical applications?
A: Torque calculation, surface normals in graphics, magnetic force on moving charges, and more.