3.The lines $L _ { 1 }$ and $L _ { 2 }$ have the equations

$$L _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 
1 \\
0 \\
9
\end{array} \right) + s \left( \begin{array} { l } 
2 \\
p \\
6
\end{array} \right) \quad \text { and } \quad L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 
- 15 \\
12 \\
- 9
\end{array} \right) + t \left( \begin{array} { r } 
4 \\
- 5 \\
2
\end{array} \right)$$

where $p$ is a constant.\\
The acute angle between $L _ { 1 }$ and $L _ { 2 }$ is $\theta$ where $\cos \theta = \frac { \sqrt { 5 } } { 3 }$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $p$ .

The line $L _ { 3 }$ has equation $\mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + u \left( \begin{array} { r } 8 \\ - 6 \\ - 5 \end{array} \right)$ and the lines $L _ { 3 }$ and $L _ { 2 }$ intersect at the point $A$ .\\
The point $B$ on $L _ { 2 }$ has position vector $\left( \begin{array} { r } 5 \\ - 13 \\ 1 \end{array} \right)$ and point $C$ lies on $L _ { 3 }$ such that $A B D C$ is a rhombus.
\item Find the two possible position vectors of $D$ .
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2018 Q3 [10]}}