| Version 6 |
Version 5 |
| \begin{defn} A \emph{rotation matrix} is a |
\begin{defn} A \emph{rotation matrix} is a |
| orthogonal matrix whose determinant is $+1$. |
orthogonal matrix whose determinant is $+1$. |
| \end{defn} |
\end{defn} |
|
|
| \subsubsection*{Examples} |
\subsubsection*{Examples} |
| \begin{enumerate} |
\begin{enumerate} |
| \item The identity matrix in $\R^n$ is a rotation matrix. |
\item The identity matrix in $\R^n$ is a rotation matrix. |
| \item The most general rotation matrix in $\R^2$ can be written as |
\item The most general rotation matrix in $\R^2$ can be written as |
| $$ |
$$ |
| \begin{pmatrix} |
\begin{pmatrix} |
| \cos \theta & \sin \theta \\ |
\cos \theta & \sin \theta \\ |
| -\sin \theta & \cos \theta |
-\sin \theta & \cos \theta |
| \end{pmatrix}, |
\end{pmatrix}, |
| $$ |
$$ |
| where $\theta\in \R$. |
where $\theta\in \R$. |
| \end{enumerate} |
\end{enumerate} |
|
|
| \subsubsection*{Properties} |
\subsubsection*{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item Rotation matrices form a group under matrix multiplication. |
\item Rotation matrices form a group under matrix multiplication. |
| In $\R^n$, this group is denoted by $\operatorname{SO}(n)$. |
In $\R^n$, this group is denoted by $\operatorname{SO}(n)$. |
| \item Suppose $v\in \R^n$ is a unit vector. |
\item Suppose $v\in \R^n$. Then there exists a rotation matrix $R$ |
| Then there exists a rotation matrix $R$ |
|
| such that $Rv = (1,0,\ldots, 0)$. |
such that $Rv = (1,0,\ldots, 0)$. |
| \end{enumerate} |
\end{enumerate} |
