Cross Product Two Vectors Calculator

Cross Product Formula:

\[ A \times B = (A_y \times B_z - A_z \times B_y,\; A_z \times B_x - A_x \times B_z,\; A_x \times B_y - A_y \times B_x) \]

Vector A (x component):

Vector A (y component):

Vector A (z component):

Vector B (x component):

Vector B (y component):

Vector B (z component):

Cross Product Result (A × B):

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1. What is 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:

\[ A \times B = (A_y \times B_z - A_z \times B_y,\; A_z \times B_x - A_x \times B_z,\; A_x \times B_y - A_y \times B_x) \]

Where:

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

Explanation: Each component of the resulting vector is calculated using the determinant of a 2×2 matrix formed by excluding the corresponding row from the 3×3 matrix of unit vectors and vector components.

3. Applications of Cross Product

Details: Cross products are essential in physics (torque, angular momentum), computer graphics (surface normals), engineering (moment of forces), and electromagnetic theory (Lorentz force).

4. Using the Calculator

Tips: Enter all six components (x, y, z for both vectors). The calculator will compute the resulting vector components. Results can be positive, negative, or zero.

5. Frequently Asked Questions (FAQ)

Q1: What does a zero cross product mean?
A: A zero cross product indicates that the vectors are parallel (or at least one is a zero vector).

Q2: How is cross product different from dot product?
A: Cross product yields a vector perpendicular to both inputs, while dot product yields a scalar representing their parallel component.

Q3: Why is cross product only defined in 3D?
A: The perpendicular vector concept only works uniquely in 3D. In 2D, it's a scalar, and higher dimensions have more complex generalizations.

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: Can I calculate cross product for 2D vectors?
A: For 2D vectors (A,B) and (C,D), treat as (A,B,0) and (C,D,0) - result will be (0,0,AD-BC).