5.The point $A$ has position vector $7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }$ and the point $B$ has position vector $12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }$ .
\begin{enumerate}[label=(\alph*)]
\item Find a vector for the line $L _ { 1 }$ which passes through $A$ and $B$ .

The line $L _ { 2 }$ has vector equation

$$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$
\item Show that $L _ { 2 }$ passes through the origin $O$ .
\item Show that $L _ { 1 }$ and $L _ { 2 }$ intersect at a point $C$ and find the position vector of $C$ .
\item Find the cosine of $\angle O C A$ .
\item Hence,or otherwise,find the shortest distance from $O$ to $L _ { 1 }$ .
\item Show that $| \overrightarrow { C O } | = | \overrightarrow { A B } |$ .
\item Find a vector equation for the line which bisects $\angle O C A$ .\\
\includegraphics[max width=\textwidth, alt={}, center]{f9d3e02c-cef2-435b-9cda-76c43fcac575-4_922_1054_279_586}

Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$, where

$$\mathrm { f } ( x ) = x \left( 12 - x ^ { 2 } \right) .$$

The curve cuts the $x$-axis at the points $P , O$ and $R$, and $Q$ is the maximum point.
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2005 Q5 [19]}}