Cross Product Calculator
A × B for 3D.
3D cross product.
How the Math Works
The cross product of two 3D vectors A and B, denoted A × B, produces a new vector perpendicular to both original vectors. Its magnitude equals the product of the vectors' lengths and the sine of the angle between them (|A||B|sinθ), representing the parallelogram area they form. The direction follows the right-hand rule: align your index finger along A, middle finger along B, and your thumb points in the cross product's direction. Mathematically, if A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), their cross product is calculated using component-wise operations involving determinants of a matrix with unit vectors i, j, k.
Practical Applications
Cross products are essential in physics for calculating torque (rotational force), magnetic forces on moving charges, and angular momentum. Engineers use them to analyze moments and structural loads in 3D systems. In computer graphics, they determine surface normals for lighting calculations, enabling realistic rendering of 3D objects. They also help compute volumes of 3D shapes (parallelepipeds) and find perpendicular vectors for coordinate system transformations in robotics and aerospace navigation.
Day-to-Day Use
Cross products appear in everyday technology like video game physics engines, where they simulate realistic collisions and movements. They're used in GPS systems to calculate orientation and in smartphones' sensors to determine device rotation. Even in sports, understanding cross products helps analyze trajectories and forces in activities like golf or tennis. Additionally, they assist in everyday tasks like determining the optimal angle for solar panels to maximize sunlight exposure throughout the day.
FAQ
Use?
Torque, normal vectors.