2010 HCI P1 Q11 [Modified]

The diagram above shows part of the structure of a modern art museum designed by Marcus, with a horizontal base \(OAB\) and vertical wall \(OADC\). Perpendicular unit vectors, \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are such that \(\mathbf{i}\) and \(\mathbf{k}\) are parallel to \(OA\) and \(OC\) respectively.

The walls of the museum \(BCD\) and \(ABD\) can be described respectively by the equations

\(\mathbf{r}\cdot \left( \begin{matrix} -1 \\ -5 \\ 6 \\ \end{matrix} \right)=36\) and \(\mathbf{r}=\left( \begin{matrix} 14 \\ 0 \\ 0 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 5 \\ 4 \\ 0 \\ \end{matrix} \right)+\mu \left( \begin{matrix} -1 \\ 0 \\ 4 \\ \end{matrix} \right)\), where \(\lambda \), \(\mu \in \mathbb{R}\).

1. Write down the distance of \(A\) from \(O\).[1]
2. Find the vector equation of the intersection line of the two walls \(BCD\) and \(ABD\).[3]
3. Marcus wishes to repaint the inner wall \(ABD\). Find the area of this wall.[3]

Suppose Marcus wishes to divide the structure into two by adding a partition such that it intersects with the walls \(BCD\) and \(ABD\) at a line. This partition can be described by the equation \(2x-y+\alpha z=\beta \), where \(\alpha \), \(\beta \in \mathbb{R}\).

4. Find the values of \(\alpha \) and \(\beta \).[2]