2022 YIJC Promo Q11

The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base \(OABC\), a vertical rectangular wall \(OADE\) and an inclined rectangular wall \(DEFG\). With respect to the origin \(O\), the vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) each of length \(1\) m, are parallel to \(OA\), \(OC\) and \(OE\) respectively. It is given that \(OE=10\)m.

The plane \(OADE\) has equation \(\mathbf{r}\cdot \left( \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right)=p\), and the plane \(DEFG\) has equation \(\mathbf{r}=\left( \begin{matrix} 0 \\ 0 \\ 10 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 3 \\ -2 \\ 4 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 0 \\ -1 \\ q \\ \end{matrix} \right)\), where \(p\) and \(q\) are constants, \(\lambda \) and \(\mu \) are parameters.

1. Give a reason why \(p=0\).[1]
2. Write down a vector equation of the line passing through the points \(D\) and \(E\).[1]
3. Show that \(q=2\).[2]
4. Hence, find the obtuse angle between the walls \(OADE\) and \(DEFG\).[3]

A rock climber is at the point \(H\left( 2,\,-1,\,12 \right)\).

5. Show that the climber is on the plane \(DEFG\).[2]

The line \(l\) passing through the points \(F\) and \(G\) has equation \(\mathbf{r}=\left( \begin{matrix} 3 \\ -2 \\ 14 \\ \end{matrix} \right)+\gamma \left( \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right)\), where \(\gamma \) is a parameter.

6. Find the exact shortest distance from the climber to \(l\).[3]