1. What is the 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. Its magnitude relates to the area of the parallelogram formed by the two vectors.

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

3. Geometric Interpretation

Details: The resulting vector is perpendicular to both input vectors, with magnitude equal to the area of the parallelogram formed by the two vectors. The direction follows the right-hand rule.

4. Using the Calculator

Tips: Enter the x, y, z components of both vectors. The calculator will compute the cross product vector components.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar result (number) while cross product gives a vector result. Dot product measures parallelness, cross product measures perpendicularity.

Q2: Can I calculate cross product in 2D?
A: Technically no, but you can treat 2D vectors as having z=0 in 3D space, which gives a result purely in the z-direction.

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

Q4: How is cross product used in physics?
A: It's used in torque calculations, magnetic force (Lorentz force), angular momentum, and more.

Q5: Why is the result perpendicular?
A: This is by definition - the cross product was designed to find a vector orthogonal to two given vectors in 3D space.