(i) The coordinates of P are \((2, 0, 0)\) since it is the midpoint of OA. The coordinates of Q are \((0, 2, 4)\) since it is the midpoint of DG. The coordinates of R are \((2, 2, 2)\) since it is the center of the face ABFE.

\(\overrightarrow{PR} = (2 - 2)\mathbf{i} + (2 - 0)\mathbf{j} + (2 - 0)\mathbf{k} = 2\mathbf{j} + 2\mathbf{k}\)

\(\overrightarrow{PQ} = (0 - 2)\mathbf{i} + (2 - 0)\mathbf{j} + (4 - 0)\mathbf{k} = -2\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\)

(ii) The dot product \(\overrightarrow{PQ} \cdot \overrightarrow{PR} = (-2)(0) + (2)(2) + (4)(2) = 8\)

The magnitudes are \(|\overrightarrow{PQ}| = \sqrt{(-2)^2 + 2^2 + 4^2} = \sqrt{24}\) and \(|\overrightarrow{PR}| = \sqrt{0^2 + 2^2 + 2^2} = \sqrt{8}\).

Using the dot product formula, \(\cos \theta = \frac{\overrightarrow{PQ} \cdot \overrightarrow{PR}}{|\overrightarrow{PQ}| |\overrightarrow{PR}|} = \frac{8}{\sqrt{24} \cdot \sqrt{8}}\)

\(\theta = \cos^{-1}\left(\frac{8}{\sqrt{24} \cdot \sqrt{8}}\right) \approx 61.9^\circ\)

(iii) \(\overrightarrow{QR} = (2 - 0)\mathbf{i} + (2 - 2)\mathbf{j} + (2 - 4)\mathbf{k} = 2\mathbf{i} - 2\mathbf{k}\)

\(|\overrightarrow{QR}| = \sqrt{2^2 + 0^2 + (-2)^2} = \sqrt{8}\)

Perimeter of \(\triangle PQR = |\overrightarrow{PR}| + |\overrightarrow{PQ}| + |\overrightarrow{QR}| = \sqrt{8} + \sqrt{24} + \sqrt{8} \approx 12.8 \text{ cm}\)