(i) To find \(\overrightarrow{AB}\), we calculate \(\overrightarrow{OB} - \overrightarrow{OA} = (4\mathbf{i} + 6\mathbf{k}) - (\mathbf{i} + 2\mathbf{j}) = 3\mathbf{i} - 2\mathbf{j} + 6\mathbf{k}\).

The magnitude of \(\overrightarrow{AB}\) is \(\sqrt{3^2 + (-2)^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7\).

Thus, the unit vector in the direction of \(\overrightarrow{AB}\) is \(\frac{3\mathbf{i} - 2\mathbf{j} + 6\mathbf{k}}{7}\).

(ii) The scalar product \(\overrightarrow{OA} \cdot \overrightarrow{OB} = 1 \times 4 + 2 \times 0 + 0 \times p = 4\).

The magnitudes are \(|\overrightarrow{OA}| = \sqrt{1^2 + 2^2} = \sqrt{5}\) and \(|\overrightarrow{OB}| = \sqrt{4^2 + p^2} = \sqrt{16 + p^2}\).

Using the cosine formula, \(4 = \sqrt{5} \times \sqrt{16 + p^2} \times \frac{1}{5}\).

Solving for \(p\), we have \(4 = \frac{\sqrt{5} \times \sqrt{16 + p^2}}{5}\).

\(20 = \sqrt{5} \times \sqrt{16 + p^2}\).

\(400 = 5 \times (16 + p^2)\).

\(400 = 80 + 5p^2\).

\(320 = 5p^2\).

\(p^2 = 64\).

\(p = \pm 8\).