(i) To find angle AOB, use the dot product formula:

\(\overrightarrow{OA} \cdot \overrightarrow{OB} = (-6) + 2 + 12 = 8\)

The magnitudes are \(|\overrightarrow{OA}| = \sqrt{14}\) and \(|\overrightarrow{OB}| = \sqrt{29}\).

\(\cos AOB = \frac{8}{\sqrt{14} \sqrt{29}}\)

\(AOB = \cos^{-1}\left(\frac{8}{\sqrt{14} \sqrt{29}}\right) \approx 66.6^\circ\)

(ii) \(\overrightarrow{BC} = \overrightarrow{OB} - \overrightarrow{OC}\)

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

\(\overrightarrow{BC} = (-3\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}) - p(2\mathbf{i} + \mathbf{j} - 3\mathbf{k})\)

\(\overrightarrow{BC} = 3\mathbf{i} - 2\mathbf{j} + 4\mathbf{k} + p(2\mathbf{i} + \mathbf{j} - 3\mathbf{k})\)

(iii) For \(BC\) to be perpendicular to \(OA\), \(\overrightarrow{BC} \cdot \overrightarrow{OA} = 0\).

\((3 + 2p) \cdot 2 + (p - 2) \cdot 1 + (4 - 3p) \cdot (-3) = 0\)

\(2(3 + 2p) + (p - 2) - 3(4 - 3p) = 0\)

\(6 + 4p + p - 2 - 12 + 9p = 0\)

\(14p - 8 = 0\)

\(p = \frac{4}{7}\) or 0.571