2008 HCI P1 Q12 [Modified]

Referring to the origin \(O\), two planes \({{\pi }_{1}}\) and \({{\pi }_{2}}\) are given by

\({{\pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix} 1 \\ 2 \\ -4 \end{matrix} \right)=13\) and \({{\pi }_{2}}:\mathbf{r}\cdot \left( \begin{matrix} 1 \\ 3 \\ 3 \end{matrix} \right)=-8\).

1. Given that a point \(A\left( 1,7,-10 \right)\) lies on \({{\pi }_{2}}\), find \(\overrightarrow{OB}\) where \(B\) is the image of \(A\) when reflected in the plane \({{\pi }_{1}}\).[4]
2. Write down the Cartesian equations of both \({{\pi }_{1}}\) and \({{\pi }_{2}}\).[1]
   Find a vector equation of the line of intersection of \({{\pi }_{1}}\) and \({{\pi }_{2}}\).[1]
3. Find a vector equation of the plane which is the image of \({{\pi }_{2}}\) when \({{\pi }_{2}}\) is reflected in \({{\pi }_{1}}\).[3]