- Relative to a fixed origin \(O\),
- the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }\)
- the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\)
- the point \(P\) has position vector \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\)
The straight line \(l\) passes through \(A\) and \(B\).
- Find a vector equation for \(l\).
The point \(C\) lies on \(l\) so that \(P C\) is perpendicular to \(l\).
- Find the coordinates of \(C\).
The point \(P ^ { \prime }\) is the reflection of \(P\) in the line \(l\).
- Find the coordinates of \(P ^ { \prime }\)
- Hence find \(\left| \overrightarrow { P P ^ { \prime } } \right|\), giving your answer as a simplified surd.