4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }\), and the point \(B\) has position vector \(7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }\).
- Find the vector \(\overrightarrow { A B }\).
- Calculate the cosine of \(\angle O A B\).
- Show that, for all values of \(\lambda\), the point P with position vector \(\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }\) lies on the line through \(A\) and \(B\).
- Find the value of \(\lambda\) for which \(O P\) is perpendicular to \(A B\).
- Hence find the coordinates of the foot of the perpendicular from \(O\) to \(A B\).