- Relative to a fixed origin \(O\),
- the point \(A\) has position vector \(\quad \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k }\)
- the point \(B\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\)
- the point \(C\) has position vector \(3 \mathbf { i } + p \mathbf { j } - \mathbf { k }\)
where \(p\) is a constant.
The line \(l\) passes through \(A\) and \(B\).
- Find a vector equation for the line \(l\)
Given that \(\overrightarrow { A C }\) is perpendicular to \(l\)
find the value of \(p\)Hence find the area of triangle \(A B C\), giving your answer as a surd in simplest form.