Editing Rotation matrix (section)

== Geometry ==
In [[Euclidean geometry]], a rotation is an example of an [[isometry]], a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave "[[chirality|handedness]]" unchanged. In contrast, a [[translation (geometry)|translation]] moves every point, a [[reflection (geometry)|reflection]] exchanges left- and right-handed ordering, a [[glide reflection]] does both, and an [[improper rotation]] combines a change in handedness with a normal rotation.

If a fixed point is taken as the origin of a [[Cartesian coordinate system]], then every point can be given coordinates as a displacement from the origin. Thus one may work with the [[vector space]] of displacements instead of the points themselves. Now suppose {{math|(''p''<sub>1</sub>, ..., ''p<sub>n</sub>'')}} are the coordinates of the vector {{math|'''p'''}} from the origin {{mvar|O}} to point {{mvar|P}}. Choose an [[orthonormal basis]] for our coordinates; then the squared distance to {{mvar|P}}, by [[Pythagorean theorem|Pythagoras]], is
:<math> d^2(O,P) = \| \mathbf{p} \|^2 = \sum_{r=1}^n p_r^2 </math>
which can be computed using the matrix multiplication
:<math> \| \mathbf{p} \|^2 = \begin{bmatrix}p_1 \cdots p_n\end{bmatrix} \begin{bmatrix}p_1 \\ \vdots \\ p_n \end{bmatrix} = \mathbf{p}^\mathsf{T} \mathbf{p} . </math>

A geometric rotation transforms lines to lines, and preserves ratios of distances between points. From these properties it can be shown that a rotation is a [[linear transformation]] of the vectors, and thus can be written in [[matrix (mathematics)|matrix]] form, {{math|''Q'''''p'''}}. The fact that a rotation preserves, not just ratios, but distances themselves, is stated as
:<math> \mathbf{p}^\mathsf{T} \mathbf{p} = (Q \mathbf{p})^\mathsf{T} (Q \mathbf{p}) , </math>
or
:<math>\begin{align}
 \mathbf{p}^\mathsf{T}  I \mathbf{p}&{}= \left(\mathbf{p}^\mathsf{T} Q^\mathsf{T}\right) (Q \mathbf{p}) \\
                    &{}= \mathbf{p}^\mathsf{T} \left(Q^\mathsf{T} Q\right) \mathbf{p} .
\end{align}</math>
Because this equation holds for all vectors, {{math|'''p'''}}, one concludes that every rotation matrix, {{math|''Q''}}, satisfies the '''orthogonality condition''',
:<math> Q^\mathsf{T} Q = I . </math>
Rotations preserve handedness because they cannot change the ordering of the axes, which implies the '''special matrix''' condition,
:<math> \det Q = +1 . </math>
Equally important, it can be shown that any matrix satisfying these two conditions acts as a rotation.