2016 SRJC J1 CT Q7

Relative to the origin \(O\), the position vectors of \(A\) and \(B\) are \(\frac{11}{3}\mathbf{i}+4\mathbf{j}+\frac{7}{3}\mathbf{k}\) and \(2\mathbf{i}+4\mathbf{k}\) respectively. It is given that \(R\) lies on \(BA\) produced such that \(2BR=3AR\).

1. Show that \(\overrightarrow{OR}\) is given by \(7\mathbf{i}+12\mathbf{j}-\mathbf{k}\) and hence find the exact length of projection of \(\overrightarrow{OR}\) on \(\overrightarrow{OB}\).
2. Find the vector \(\overrightarrow{NR}\), where \(N\) is the foot of perpendicular from \(R\) to \(OB\).
3. Find the position vector of the point \(R'\), which is a reflection of the point \(R\) in the line passing through \(OB\).