Editing Rotation matrix (section)

== Ambiguities ==
[[File:Alias and alibi rotations.png|thumb|350px|right|Alias and alibi rotations]]
The interpretation of a rotation matrix can be subject to many ambiguities.

In most cases the effect of the ambiguity is equivalent to the effect of a rotation matrix [[Invertible matrix|inversion]] (for these orthogonal matrices equivalently matrix [[transpose]]).

; Alias or alibi (passive or active) transformation
: The coordinates of a point {{math|''P''}} may change due to either a rotation of the coordinate system {{math|''CS''}} ([[Active and passive transformation|alias]]), or a rotation of the point {{math|''P''}} ([[Active and passive transformation|alibi]]). In the latter case, the rotation of {{math|''P''}} also produces a rotation of the vector {{math|'''v'''}} representing {{math|''P''}}. In other words, either {{math|''P''}} and {{math|'''v'''}} are fixed while {{math|''CS''}} rotates (alias), or {{math|''CS''}} is fixed while {{math|''P''}} and {{math|'''v'''}} rotate (alibi). Any given rotation can be legitimately described both ways, as vectors and coordinate systems actually rotate with respect to each other, about the same axis but in opposite directions. Throughout this article, we chose the alibi approach to describe rotations. For instance, 
::<math>
R(\theta) = \begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{bmatrix}</math>
: represents a counterclockwise rotation of a vector {{math|'''v'''}} by an angle {{math|''θ''}}, or a rotation of {{math|''CS''}} by the same angle but in the opposite direction (i.e. clockwise). Alibi and alias transformations are also known as [[active and passive transformation]]s, respectively.
; Pre-multiplication or post-multiplication
: The same point {{math|''P''}} can be represented either by a [[column vector]] {{math|'''v'''}} or a [[row vector]] {{math|'''w'''}}. Rotation matrices can either pre-multiply column vectors ({{math|''R'''''v'''}}), or post-multiply row vectors ({{math|'''w'''''R''}}). However, {{math|''R'''''v'''}} produces a rotation in the opposite direction with respect to {{math|'''w'''''R''}}. Throughout this article, rotations produced on column vectors are described by means of a pre-multiplication. To obtain exactly the same rotation (i.e. the same final coordinates of point {{math|''P''}}), the equivalent row vector must be post-multiplied by the [[transpose]] of {{mvar|R}} (i.e. {{math|'''w'''''R''<sup>T</sup>}}).
; Right- or left-handed coordinates
: The matrix and the vector can be represented with respect to a [[Cartesian coordinate system#Orientation and handedness|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 form]]s, and the matrix can act on either vectors or forms.