(i) For \(\overrightarrow{OA}\) to be perpendicular to \(\overrightarrow{OB}\), their dot product must be zero:

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

\(-8 + 3 + p = 0\)

\(p = 5\)

(ii) The vector \(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}\) is given by:

\(\overrightarrow{AB} = \begin{pmatrix} 4 \\ 1 \\ p \end{pmatrix} - \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \\ p-1 \end{pmatrix}\)

The magnitude of \(\overrightarrow{AB}\) is 7:

\(\sqrt{6^2 + (-2)^2 + (p-1)^2} = 7\)

\(\sqrt{36 + 4 + (p-1)^2} = 7\)

\(36 + 4 + (p-1)^2 = 49\)

\((p-1)^2 = 9\)

\(p-1 = 3\) or \(p-1 = -3\)

\(p = 4\) or \(p = -2\)