N1993 P1 Q14

The equation of a straight line \(l\) is \(\mathbf{r}=\left( \begin{matrix} 1 \\ 2 \\ 3 \\ \end{matrix} \right)+t\left( \begin{matrix} -1 \\ 1 \\ 1 \\ \end{matrix} \right)\), where \(t\) is a parameter. The point \(A\)on \(l\) is given by \(t=0\), and the origin of position vectors is \(O\).

1. Calculate the acute angle between \(OA\) and \(\ell \), giving your answer correct to the nearest degree.[3]
2. Find the position vector of the point \(P\) on \(\ell \) such that \(OP\) is perpendicular to \(\ell \).[3]
3. A point \(Q\) on \(\ell \) is such that the length of \(OQ\) is \(5\) units. Find the two possible position vectors of \(Q\).[3]
4. The points \(R\) and \(S\) on \(\ell \) are given by \(t\,=\,\lambda \) and \(t\,=\,2\lambda \) respectively. Show that there is no value of \(\lambda \) for which \(OR\) and \(OS\) are perpendicular.[3]