(a) Calculate \(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix} - \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ -2 \\ -4 \end{pmatrix}.\)

Calculate \(\overrightarrow{CD} = \overrightarrow{OD} - \overrightarrow{OC} = \begin{pmatrix} 3 \\ 2 \\ 3 \end{pmatrix} - \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}.\)

Find the magnitudes: \(AB = \sqrt{2^2 + (-2)^2 + (-4)^2} = \sqrt{24}, \quad CD = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{6}.\)

Thus, \(AB = 2CD.\)

(b) The angle \(\theta\) between \(\overrightarrow{AB}\) and \(\overrightarrow{CD}\) is given by \(\cos \theta = \frac{\overrightarrow{AB} \cdot \overrightarrow{CD}}{|\overrightarrow{AB}| |\overrightarrow{CD}|}.\)

Calculate the dot product: \(\overrightarrow{AB} \cdot \overrightarrow{CD} = 2 \times 2 + (-2) \times 1 + (-4) \times 1 = 2.\)

Thus, \(\cos \theta = \frac{2}{\sqrt{24} \times \sqrt{6}} = \frac{2}{12} = \frac{1}{6}.\)

Therefore, \(\theta = \cos^{-1} \left( \frac{1}{6} \right) \approx 99.6^\circ.\)

(c) The vector equations of the lines are:

Line through A and B: \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ -4 \end{pmatrix}.\)

Line through C and D: \(\mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}.\)

Equate components and solve for \(\lambda\) and \(\mu\):

\(2 + 2\lambda = 1 + 2\mu, \quad 1 - 2\lambda = 1 + \mu, \quad 5 - 4\lambda = 2 + \mu.\)

Solving these gives inconsistent results, confirming the lines do not intersect.