1. \(\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about }
   \text { the } z \text {-axis is represented by the matrix }
   \qquad \left( \begin{array} { c c c } \cos \theta & - \sin \theta & 0
   \sin \theta & \cos \theta & 0
   0 & 0 & 1 \end{array} \right) \end{array} \right]\)

Given that the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { c c c } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } & 0
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } & 0
0 & 0 & 1 \end{array} \right)$$ represents a rotation through \(\alpha ^ { \circ }\) anticlockwise about the \(z\)-axis with respect to the right-hand rule,

1. determine the value of \(\alpha\).
2. Hence determine the smallest possible positive integer value of \(k\) for which \(\mathbf { M } ^ { k } = \mathbf { I }\) The \(3 \times 3\) matrix \(\mathbf { N }\) represents a reflection in the plane with equation \(y = 0\)
3. Write down the matrix \(\mathbf { N }\). The point \(A\) has coordinates (-2, 4, 3)
   The point \(B\) is the image of the point \(A\) under the transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\).
4. Show that the coordinates of \(B\) are \(( 2 + \sqrt { 3 } , 2 \sqrt { 3 } - 1,3 )\) Given that \(O\) is the origin,
5. show that, to 3 significant figures, the size of angle \(A O B\) is \(66.9 ^ { \circ }\)
6. Hence determine the area of triangle \(A O B\), giving your answer to 3 significant figures.