(i) Express the general point of l or m in component form:

For l: \((2 + s, -1 + s, 4 - s)\)

For m: \((-2 - 2t, 2 + t, 1 + t)\)

Equate at least two pairs of components and solve for \(s\) or \(t\).

Possible solutions: \(s = \frac{2}{3}, 10, 3\) or \(t = \frac{7}{3}, -7, 0\).

Verify that all three component equations are not satisfied, hence lines do not intersect.

(ii) Express \(\overrightarrow{PQ}\) in terms of \(s\):

\(\overrightarrow{PQ} = -s\mathbf{i} + (1 - s)\mathbf{j} + (-5 + s)\mathbf{k}\)

Equate its scalar product with a direction vector for l to zero:

\((-s)(1) + (1-s)(1) + (-5+s)(-1) = 0\)

Solve for \(s\): \(s = 2\)

\(\mathbf{OP} = 4\mathbf{i} + \mathbf{j} + 2\mathbf{k}\)

(iii) Show that Q is on m with parameter \(t = -2\):

Substitute \(t = -2\) into m's equation:

\(\mathbf{r} = -2\mathbf{i} + 2\mathbf{j} + \mathbf{k} + (-2)(-2\mathbf{i} + \mathbf{j} + \mathbf{k}) = 2\mathbf{i} - \mathbf{k}\)

Verify \(PQ\) is perpendicular to m:

\((-2\mathbf{i} - \mathbf{j} - 3\mathbf{k}) \cdot (-2\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0\)