(a) To find the vector equation for the line passing through A and B, use the position vectors of A and B:

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

Equate the components of the general points on their line and \(l\):

\(\begin{align*} 1 + \lambda &= 1 + 2\mu, \\ 2 - 3\lambda &= -1 - 3\mu, \\ -2 + 3\lambda &= 3 + 4\mu. \end{align*}\)

Solving these equations gives \(\lambda = -1\) and \(\mu = -2\). However, substituting these values into the third equation does not satisfy it, confirming the lines do not intersect.

(b) For a general point \(P\) on \(l\), \(\mathbf{P} = -3\mathbf{j} + 5\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k})\).

Calculate the scalar product of \(\overrightarrow{AP}\) and a direction vector for \(l\) and equate to zero:

\(4\mu + (9 + 9\mu) + (20 + 16\mu) = 0\)

Solving gives \(\mu = -1\).

Substitute \(\mu = -1\) back into the equation for \(P\) to find the position vector:

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