10 The line $l _ { 1 }$ is parallel to the vector $\mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }$ and passes through the point $A$, whose position vector is $3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$. The line $l _ { 2 }$ is parallel to the vector $- 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }$ and passes through the point $B$, whose position vector is $- 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$. The point $P$ on $l _ { 1 }$ and the point $Q$ on $l _ { 2 }$ are such that $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$. Find\\
(i) the length $P Q$,\\
(ii) the cartesian equation of the plane $\Pi$ containing $P Q$ and $l _ { 2 }$,\\
(iii) the perpendicular distance of $A$ from $\Pi$.

\hfill \mbox{\textit{CAIE FP1 2014 Q10}}