2022 HCI J2 BT Q7

Taking the point \(O\) as the origin, the diagram above shows a right trapezoidal prism that is made up of \(2\)congruent trapeziums \(OAFG\) and \(BCDE\) that are joined by \(4\) rectangles \(OABC\), \(DEFG\), \(ABEF\) and \(OCDG\). Perpendicular unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are such that \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OA\) and \(OC\) respectively. The length of \(OA\) is \(8\) units, the length of \(AB\) is \(10\) units and the length of \(GF\) is \(4\) units. The \(2\) rectangles \(ABEF\) and \(OCDG\) are inclined to the horizontal plane \(OABC\) at an angle of \(45{}^\circ \).

1. Show that the equation of the plane \(ABEF\) may be expressed as \(x+z=8\).[2]
2. Given that the equation of the plane \(OCDG\) is \(\mathbf{r}\cdot \left( \begin{matrix} -1 \\ 0 \\ 1 \\ \end{matrix} \right)=k\), where \(k\) is a constant to be determined, find the vector equation of the line \(l\) where planes \(ABEF\) and \(OCDG\) meet. Hence state the distance from \(l\) to the plane \(DEFG\).[3]
3. The reflection of  \(l\) in the plane \(DEFG\) is \(l'\). Write down the Cartesian equation of \(l'\).[1]
4. The line \(m\), which passes through the point \(H\) has equation \(x-1=\frac{y+2}{3}=z\). Find the possible coordinates of \(H\) such that the shortest distance from \(H\) to the planes \(ABEF\) and \(OCDG\) is the same.[4]
5. Determine if the point of intersection between the line \(m\) and the plane \(ABEF\) lies on the rectangular surface \(ABEF\). Justify your answer.[2]