One arrow, two descriptions

Rotate the basis. Keep the vector.

Coordinates belong to a basis. Compute the new column, rebuild the same arrow, then see exactly what breaks when you forget to rotate the basis too.

AT 180°, DO NEGATIVE COMPONENTS REVERSE THE PHYSICAL ARROW?

VECTOR X

VECTOR Y

BASIS ROTATION · 180°

R and Rᵀ will appear after computation.

Coordinate field ready

READY

ROTATED COORDINATES-

VECTOR NORM-

ENDPOINT ERROR-

Coordinate transform

The dot products with the rotated basis vectors produce the coordinate column.

Reconstruction

The transformed coordinates and transformed basis recombine to the original endpoint.

Invariant

Orthogonal rotations preserve inner products, lengths, and angles.

2D orthonormal Euclidean bases only. This lab does not model non-orthogonal coordinates, curved spaces, or higher-rank tensor transformation laws.

Awaiting computation