5．The point \(A\) has position vector \(7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }\) and the point \(B\) has position vector \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) ．
（a）Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) ． The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$ （b）Show that \(L _ { 2 }\) passes through the origin \(O\) ．
（c）Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) ．
（d）Find the cosine of \(\angle O C A\) ．
（e）Hence，or otherwise，find the shortest distance from \(O\) to \(L _ { 1 }\) ．
（f）Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) ．
（g）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.