To find the vector equation of the line passing through points A and B, we use the position vectors:

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

\(\overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}\)

The direction vector \(\overrightarrow{AB}\) is given by:

\(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = (3\mathbf{i} + \mathbf{j} + \mathbf{k}) - (\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})\)

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

The vector equation of the line through A and B is:

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

Equating the components of the line \(l\) and the line through A and B:

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

\(1 - 2\mu = -2 + 3\lambda\)

\(m - 4\mu = 2 - \lambda\)

Solving these equations, we find \(\lambda = \frac{5}{7}\) and \(\mu = \frac{3}{7}\).

Substituting \(\mu = \frac{3}{7}\) into the third equation:

\(m - 4\left(\frac{3}{7}\right) = 2 - \frac{5}{7}\)

\(m - \frac{12}{7} = \frac{9}{7}\)

\(m = \frac{9}{7} + \frac{12}{7} = 3\)

Thus, the value of \(m\) is \(3\).