Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, $l_1$ and $l_2$, along which they travel are
\begin{align}
\mathbf{r} &= 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) \\
\text{and} \quad \mathbf{r} &= 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k}),
\end{align}
where $\lambda$ and $\mu$ are scalars.

\begin{enumerate}[label=(\alph*)]
\item Show that the submarines are moving in perpendicular directions. [2]
\item Given that $l_1$ and $l_2$ intersect at the point $A$, find the position vector of $A$. [5]
\end{enumerate}

The point $B$ has position vector $10\mathbf{j} - 11\mathbf{k}$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that only one of the submarines passes through the point $B$. [3]
\item Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance $AB$. [2]
\end{enumerate}

\hfill \mbox{\textit{Edexcel C4  Q7 [12]}}