🧠Vectors are Independent of Basis, Components Transform via Rotation Matrices

While the numerical components of a vector change when switching between different orthonormal bases, the vector itself (its physical direction and magnitude) remains invariant. This transformation between orthonormal bases is described by a rotation matrix, whose elements are the dot products of the old and new basis vectors. These transformation coefficients ensure that fundamental properties like vector magnitude and the scalar product (angle between vectors) are preserved, highlighting their role as invariants in physics.

This section demonstrates rotations and basis changes in cloud computing, offering both an animated web visualization of 2D base rotations and vector transformations, alongside a numerical analysis of vector component changes in a new basis.

🎬Animated result and interactive web

Rotated 2D Bases Visualization

Vector Transformation between bases