Solve perpendicularity conditions and compare two area methods

The points $A$, $B$, $C$ and $D$ have position vectors $\mathbf i-2\mathbf j+5\mathbf k$, $\mathbf i+3\mathbf j$, $10\mathbf i+\mathbf j+2\mathbf k$ and $-2\mathbf i+4\mathbf j+5\mathbf k$ respectively, with respect to an origin $O$. The point $P$ on $AB$ is such that $AP:PB=\lambda:1-\lambda$ and the point $Q$ on $CD$ is such that $CQ:QD=\mu:1-\mu$. Find $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ in terms of $\lambda$ and $\mu$ respectively.

Given that $\overrightarrow{PQ}$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{CD}$,

(i) show that $\overrightarrow{PQ}=\mathbf i+2\mathbf j+2\mathbf k$,

(ii) find the area of the triangle $ABQ$.