# 14.1: C.1- Infintesimal Rotations

- Page ID
- 18091

Any rotation matrix can be written as the identity plus something:

\[\underset{\sim}{C}=\underset{\sim}{\delta}+\underset{\sim}{r}\]

The orthogonality requirement \(\underset{\sim}{C}^T \underset{\sim}{C} = \underset{\sim}{\delta}\) can then be expressed as:

\[\begin{aligned}

\underset{\sim}{C}^{T} \underset{\sim}{C} &=\left(\underset{\sim}{\delta}^{T}+\underset{\sim}{r}^{T}\right)(\underset{\sim}{Q}+\underset{\sim}{r}) \\

&=\underset{\sim}{\delta}^{T} \underset{\sim}{\delta}+\underset{\sim}{\delta}^{T} \underset{\sim}{r}+\underset{\sim}{\delta} \underset{\sim}{r}^{T}+\underset{\sim}{r}^{T} \underset{\sim}{r} \\

&=\underset{\sim}{\delta}+ \underset{\sim}{\delta} \underset{\sim}{r}+\underset{\sim}{\delta} \underset{\sim}{r}^{T}+\underset{\sim}{r}^{T} \underset{\sim}{r}=\underset{\sim}{\delta},

\end{aligned}\]

(using the identities \(\underset{\sim}{\delta}^T\) = \(\underset{\sim}{\delta}\) and \(\underset{\sim}{\delta}\underset{\sim}{\delta}\) = \(\underset{\sim}{\delta}\)), and therefore

\[\underset{\sim}{r}+\underset{\sim}{r}^{T}+\underset{\sim}{r}^{T} \underset{\sim}{r}=0.\label{eqn:1}\]

Now suppose that the rotation is through a very small angle^{1}. In this case, all components of \(\underset{\sim}{r}\) are \(\ll 1\), and the third term on the left-hand side of Equation \(\ref{eqn:1}\) is therefore negligible, leaving us with

\[\underset{\sim}{r}+\underset{\sim}{r}^{T}=0.\]

So, for an infinitesimal rotation, the rotation matrix equals the identity plus an *antisymmetric* matrix whose elements are \(\ll 1\).

^{1}More precisely, we take the limit as the angle goes to zero.