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.
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