# Definition The [[Orthogonal Group]] in 3 dimensions, $O(3)$ is a [[Lie group]] consisting of all proper and improper [[Rotation|rotations]] in 3 spatial dimensions. It is a [[Connected Group|disconnected group]]. ![[Pasted image 20210703131553.png]] # Proper Rotations Proper rotations are the elements of $SO(3)$ and are described on the page [[Special Orthogonal Group in 3 Dimensions]]. # Improper Rotations Improper rotations are characterized by $\det(R) = -1$. They can be decomposed as an inversion and a proper [[Rotation]], where the [[inversion operator]] is represented by: $ P \equiv -I = \begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix} $