N2012 P1 Q9

Find a vector equation of the line through the points \(A\) and \(B\) with position vectors \(\mathrm{7}\mathbf{i}\,+\,8\mathbf{j}\,+\,9\mathbf{k}\) and \(-\mathbf{i}-8\mathbf{j}+\mathbf{k}\) respectively.[3]

The perpendicular to this line from the point \(C\) with position vector \(\mathbf{i}+8\mathbf{j}+3\mathbf{k}\) meets the line at point \(N\). Find the position vector of \(N\) and the ratio \(AN:NB\).[5]

Find the Cartesian equation of the line which is a reflection of the line \(AC\) in the line \(AB\).[4]