(i) The vector equation of the line through A and B is \(\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - \mathbf{j} + 2\mathbf{k})\).

Equating components with the line \(l\):

\(1 + \lambda = 2 - \mu\)

\(1 - \lambda = -2 + 2\mu\)

\(1 + 2\lambda = -1 + \mu\)

Solving these equations, we find \(\lambda = -1\) and \(\mu = 2\), but the third equation is not satisfied, showing no intersection.

(ii) To find the perpendicular distance from A to \(l\), consider a point \(P\) on \(l\) given by \((1-\mu)\mathbf{i} + (-3 + 2\mu)\mathbf{j} + (-2 + \mu)\mathbf{k}\).

Calculate \(\overrightarrow{AP}\) and find \(\mu\) such that \(\overrightarrow{AP}\) is perpendicular to the direction vector of \(l\), \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\).

Solving gives \(\mu = \frac{3}{2}\).

Substitute \(\mu = \frac{3}{2}\) into \(\overrightarrow{AP}\) and find its magnitude: \(\frac{1}{\sqrt{2}}\).