7. With respect to a fixed origin \(O\),
- the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4
2
- 3 \end{array} \right) + \lambda \left( \begin{array} { r } - 4
- 3
5 \end{array} \right)\) where \(\lambda\) is a scalar constant - the point \(A\) has position vector \(9 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\)
Given that \(X\) is the point on \(l\) nearest to \(A\),
- find
- the coordinates of \(X\)
- the shortest distance from \(A\) to \(l\).
Give your answer in the form \(\sqrt { d }\), where \(d\) is an integer.
The point \(B\) is the image of \(A\) after reflection in \(l\).
- Find the position vector of \(B\).
Solutions relying on calculator technology are not acceptable.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-26_668_661_408_644}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}