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 equals the area of the parallelogram spanned by the two vectors.

2. How Does The Calculator Work?

The calculator uses the cross product magnitude formula:

Where:

• \( |\mathbf{A}| \) — Magnitude of vector A
• \( |\mathbf{B}| \) — Magnitude of vector B
• \( \theta \) — Angle between vectors A and B (in degrees)
• \( \sin(\theta) \) — Sine of the angle between vectors

Explanation: The cross product magnitude represents the area of the parallelogram formed by the two vectors and is maximum when vectors are perpendicular.

3. Applications Of Cross Product

Details: Cross products are essential in physics for calculating torque, angular momentum, magnetic force, and in computer graphics for surface normals and lighting calculations.

4. Using The Calculator

Tips: Enter positive magnitudes for both vectors and an angle between 0° and 180°. The result gives the magnitude of the cross product vector.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between cross product and dot product?
A: Cross product gives a vector (perpendicular to both inputs) while dot product gives a scalar (related to cosine of angle).

Q2: When is the cross product zero?
A: When vectors are parallel (angle = 0° or 180°) or when either vector has zero magnitude.

Q3: What is the maximum possible cross product magnitude?
A: Maximum occurs when vectors are perpendicular (θ = 90°), giving |A| × |B|.

Q4: Can cross product be calculated in 2D?
A: In 2D, cross product gives a scalar representing the signed area, while in 3D it gives a vector.

Q5: What are the units of cross product magnitude?
A: Units are the square of the input vector units (e.g., m² if vectors are in meters).