Relative to a fixed origin $O$, the point $A$ has position vector $\begin{pmatrix} -2 \\ 4 \\ 7 \end{pmatrix}$

and the point $B$ has position vector $\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix}$

The line $l_1$ passes through the points $A$ and $B$.

\begin{enumerate}
\item[(a)] Find the vector $\overrightarrow{AB}$. \hfill [2]

\item[(b)] Hence find a vector equation for the line $l_1$ \hfill [1]
\end{enumerate}

The point $P$ has position vector $\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}$

Given that angle $PBA$ is $\theta$,

\begin{enumerate}
\item[(c)] show that $\cos\theta = \frac{1}{3}$ \hfill [3]
\end{enumerate}

The line $l_2$ passes through the point $P$ and is parallel to the line $l_1$

\begin{enumerate}
\item[(d)] Find a vector equation for the line $l_2$ \hfill [2]
\end{enumerate}

The points $C$ and $D$ both lie on the line $l_2$

Given that $AB = PC = DP$ and the $x$ coordinate of $C$ is positive,

\begin{enumerate}
\item[(e)] find the coordinates of $C$ and the coordinates of $D$. \hfill [3]

\item[(f)] find the exact area of the trapezium $ABCD$, giving your answer as a simplified surd. \hfill [4]
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{Edexcel C4 2014 Q8 [15]}}