(i) To find \(k\) when \(\angle AOB = 90^\circ\), use the scalar product: \(\overrightarrow{OA} \cdot \overrightarrow{OB} = 0\).

\(\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} \cdot \begin{pmatrix} 5 \\ -1 \\ k \end{pmatrix} = 2 \times 5 + 1 \times (-1) + (-2) \times k = 0\)

\(10 - 1 - 2k = 0\)

\(9 = 2k\)

\(k = \frac{9}{2}\)

(ii) For \(|\overrightarrow{AB}| = |\overrightarrow{OC}|\), first find \(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = \begin{pmatrix} 5 \\ -1 \\ k \end{pmatrix} - \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ k+2 \end{pmatrix}\).

\(|\overrightarrow{OC}| = \sqrt{2^2 + 6^2 + (-3)^2} = 7\).

Equate magnitudes: \(\sqrt{3^2 + (-2)^2 + (k+2)^2} = 7\).

\(9 + 4 + (k+2)^2 = 49\)

\((k+2)^2 = 36\)

\(k+2 = 6\) or \(k+2 = -6\)

\(k = 4\) or \(k = -8\)

(iii) \(\overrightarrow{OD} = 3 \overrightarrow{OA} = 3 \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} = \begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\).

\(\overrightarrow{OE} = 2 \overrightarrow{OC} = 2 \begin{pmatrix} 2 \\ 6 \\ -3 \end{pmatrix} = \begin{pmatrix} 4 \\ 12 \\ -6 \end{pmatrix}\).

\(\overrightarrow{DE} = \overrightarrow{OE} - \overrightarrow{OD} = \begin{pmatrix} 4 \\ 12 \\ -6 \end{pmatrix} - \begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix} = \begin{pmatrix} -2 \\ 9 \\ 0 \end{pmatrix}\).

Magnitude of \(\overrightarrow{DE} = \sqrt{(-2)^2 + 9^2 + 0^2} = \sqrt{4 + 81} = \sqrt{85}\).