8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where,
$$P = \left( \begin{array} { r r }
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\end{array} \right)$$
- 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.
- Write down the matrix \(\mathbf { Q }\).
Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
- 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 \(( 0,0 )\), - find the scale factor of the enlargement,
- find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.