8. (i) The transformation $U$ is represented by the matrix $\mathbf { P }$ where,

$$P = \left( \begin{array} { r r } 
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\
\frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the transformation $U$.

The transformation $V$, represented by the matrix $\mathbf { Q }$, is a stretch scale factor 3 parallel to the $x$-axis.
\item Write down the matrix $\mathbf { Q }$.

Transformation $U$ followed by transformation $V$ is a transformation which is represented by matrix $\mathbf { R }$.
\item Find the matrix $\mathbf { R }$.\\
(ii)

$$S = \left( \begin{array} { r r } 
1 & - 3 \\
3 & 1
\end{array} \right)$$

Given that the matrix $\mathbf { S }$ represents an enlargement, with a positive scale factor and centre $( 0,0 )$, followed by a rotation with centre $( 0,0 )$,\\
(a) find the scale factor of the enlargement,\\
(b) find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.
\end{enumerate}

\hfill \mbox{\textit{Edexcel F1  Q8 [11]}}