The position vectors of the points \(A , B , C , D\) are
\(7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\),
\(3 \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\),
\(2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\),
\(2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }\)
respectively. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 .
- Show that \(\lambda ^ { 2 } - 5 \lambda + 4 = 0\).
- Find the acute angle between the planes through \(A , B , D\) corresponding to the values of \(\lambda\) satisfying the equation in part (i).