10 The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5 \mathbf { i } + 2 \mathbf { j } - 9 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k }\) respectively.
- Find a vector equation for the line \(P Q\).
The position vector of the point \(T\) is \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\).
- Write down a vector equation for the line \(O T\) and show that \(O T\) is perpendicular to \(P Q\).
It is given that \(O T\) intersects \(P Q\).
- Find the position vector of the point of intersection of \(O T\) and \(P Q\).
- Hence find the perpendicular distance from \(O\) to \(P Q\), giving your answer in an exact form.