$\overrightarrow{AB}=3 \hat i-\hat j+\hat k \;and\; \overrightarrow{CD}=-3\hat i+2 \hat j+4\hat k$ are two vectors. The position vectors of the points $A$ and $C$ are $6\hat i+7\hat j+4\hat k\;and\;-9\hat j+2\hat k,$ respectively.Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{PQ}$ is perpendicular to $\overrightarrow{AB}\;and\;\overrightarrow{CD}$ both.

Question

$\begin{array}{1 1} P(3\hat i + 8\hat j + 3\hat k) \: and \: Q(-3\hat i - 7\hat j + 6\hat k ) \\ P(2\hat i + 8\hat j + 3\hat k) \: and \: Q(-2\hat i - 7\hat j + 6\hat k )\\ P(3\hat i + 7\hat j + 3\hat k) \: and \: Q(-3\hat i +7\hat j + 6\hat k ) \\P(3\hat i + 8\hat j - 3\hat k) \: and \: Q(3\hat i - 7\hat j + 6\hat k ) \end{array} $