With respect to the origin O, the points A, B, C, D have position vectors given by

\(\overrightarrow{OA} = 4\mathbf{i} + \mathbf{k}, \quad \overrightarrow{OB} = 5\mathbf{i} - 2\mathbf{j} - 2\mathbf{k}, \quad \overrightarrow{OC} = \mathbf{i} + \mathbf{j}, \quad \overrightarrow{OD} = -\mathbf{i} - 4\mathbf{k}\)

1. Calculate the acute angle between the lines AB and CD.
2. Prove that the lines AB and CD intersect.
3. The point P has position vector \(\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\). Show that the perpendicular distance from P to the line AB is equal to \(\sqrt{3}\).