Rotation Matrix - Ambiguities

Ambiguities

The interpretation of a rotation matrix can be subject to many ambiguities.

Alias or alibi transformation

The change in a vector's coordinates can be due to a turn of the coordinate system (alias) or a turn of the vector (alibi). Any rotation can be legitimately described both ways, as vectors and coordinate systems actually rotate with respect to each other. Throughout this article, we chose the alibi approach to describe rotations.

Pre-multiplication or post-multiplication

The vector can be pre-multiplied by a rotation matrix (Rv, where v is a column vector), or post-multiplied by it (vR, where v is a row vector). Throughout this article, we described rotations produced by means of a pre-multiplication.

Right- or left-handed coordinates

The matrix and the vector can be represented with respect to a right-handed or left-handed coordinate system. Throughout the article, we assumed a right-handed orientation, unless otherwise specified.

Vectors or forms

The vector space has a dual space of linear forms, and the matrix can act on either vectors or forms.

In most cases the effect of the ambiguity is equivalent to the effect of a transposition of the rotation matrix.