- The point \(A\), with coordinates \(( 0 , a , b )\) lies on the line \(l _ { 1 }\), which has equation
$$\mathbf { r } = 6 \mathbf { i } + 19 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } )$$
- Find the values of \(a\) and \(b\).
The point \(P\) lies on \(l _ { 1 }\) and is such that \(O P\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin.
- Find the position vector of point \(P\).
Given that \(B\) has coordinates \(( 5,15,1 )\),
- show that the points \(A , P\) and \(B\) are collinear and find the ratio \(A P : P B\).