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The lines \(l_1\) and \(l_2\) have equations

\(\mathbf r=8\mathbf i+2\mathbf j+3\mathbf k+\lambda(\mathbf i-2\mathbf j)\)

and

\(\mathbf r=5\mathbf i+3\mathbf j-14\mathbf k+\mu(2\mathbf j-3\mathbf k)\).

The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\). Find the position vector of \(P\) and the position vector of \(Q\).

The points with position vectors \(8\mathbf i+2\mathbf j+3\mathbf k\) and \(5\mathbf i+3\mathbf j-14\mathbf k\) are denoted by \(A\) and \(B\), respectively. Find

(i) \(\overrightarrow{AP}\times\overrightarrow{AQ}\) and hence the area of triangle \(APQ\),

(ii) the volume of the tetrahedron \(APQB\).