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.

R and Rᵀ will appear after computation.

Coordinate field ready

READY
ROTATED COORDINATES-
VECTOR NORM-
ENDPOINT ERROR-
Coordinate transform

c = Rᵀv

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

Reconstruction

v = Rc

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

Invariant

||v|| = 3.605551

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

Same vector, different basis coordinates

Read the explanation

The source starts with vector two three and a basis rotated one hundred eighty degrees. Rotated coordinates are computed with the transposed rotation matrix. They become approximately minus two minus three. Multiplying these coordinates by the rotated basis reconstructs the original two three endpoint. The visible arrow represents the reconstructed physical vector; changing its coordinate column does not by itself rotate that vector. The original vector’s Euclidean norm is the square root of two squared plus three squared: square root thirteen, about three point six zero six. The rotated coordinates minus two minus three have the same squared sum. This source checks a two dimensional orthonormal Euclidean basis. It does not establish invariance for arbitrary nonorthogonal coordinates or higher rank tensors. The original counterexample button interprets the new coordinate column in the old basis. At one hundred eighty degrees it reads minus two minus three as the endpoint instead of reconstructing two three. The separation is square root of four squared plus six squared, or square root fifty two, about seven point two one one. Matching norms alone would miss this error, since both arrows still have norm square root thirteen. The original exported reconstruction and endpoint error provide the evidence.

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