JPJC Tutorial Q6

The position vectors of \(A\) and \(B\) are \(\mathrm{2}\mathbf{i}\,\mathrm{+3}\mathbf{j}-\mathbf{k}\) and \(\mathrm{4}\mathbf{i}\,- \mathrm{2}\mathbf{j }\mathrm{+ 2}\mathbf{k}\) respectively. The line \(l\), which passed through \(A\), has equation \(\mathrm{r = (2}\mathbf{i }\mathrm{+ 3}\mathbf{j}-\mathbf{k}\mathrm{)}\,\mathrm{+}\,\mathrm{ }\!\!\mu\!\!\mathrm{ }\,\mathrm{(}\mathbf{i}\,-\,\mathrm{2}\mathbf{j}\,\,\mathrm{+}\,\mathrm{2}\mathbf{k}\mathrm{)}\), where \(\mu \) is a parameter.

1. Find the position vector of \(Q\), the foot of perpendicular from \(B\) to \(l\). Hence, find the perpendicular distance from \(B\) to \(l\).[4]
2. Find the length of projection of \(\overrightarrow{AB}\) on \(l\) and hence verify your answer for the perpendicular distance in (i).[4]