First, find the vector equation of the line passing through A and B. The direction vector \(\overrightarrow{AB}\) is given by:

\(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = (\mathbf{i} + \mathbf{j} + 5\mathbf{k}) - (2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = -\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\)

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

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

Equate the components of the general points on both lines:

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

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

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

Solve these equations for \(\lambda\) and \(\mu\):

From the first equation: \(\lambda = 1 - 3\mu\)

Substitute into the second equation:

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

\(-1 + 2 - 6\mu = 1 + \mu\)

\(-6\mu - \mu = 1\)

\(-7\mu = 0\)

\(\mu = 0\)

Substitute \(\mu = 0\) into \(\lambda = 1 - 3\mu\):

\(\lambda = 1\)

Check the third equation:

\(3 + 2(1) = 2 - 0\)

\(5 \neq 2\)

Since not all three equations are satisfied, the lines do not intersect.