(a) To find the position vector of M, the midpoint of AC, use the midpoint formula:

\(\overrightarrow{OM} = \frac{1}{2} (\overrightarrow{OA} + \overrightarrow{OC}) = \frac{1}{2} \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \begin{pmatrix} 4 \\ -3 \\ -2 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}.\)

\(For N, given BN = 2NC, we find:\)

\(\overrightarrow{ON} = \overrightarrow{OB} + \frac{2}{3}(\overrightarrow{OC} - \overrightarrow{OB}) = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \frac{2}{3} \begin{pmatrix} 3 \\ -3 \\ -3 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ -1 \end{pmatrix}.\)

(b) The vector equation for the line through M and N is:

\(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -3 \\ -1 \end{pmatrix}.\)

(c) The vector equation for the line through A and B is:

\(\mathbf{r} = \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -5 \\ -1 \end{pmatrix}.\)

Equating components of AB and MN gives:

\(\lambda = -3\) or \(\mu = 2.\)

Substituting \(\mu = 2\) into the equation for AB gives:

\(\overrightarrow{OQ} = \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + 2 \begin{pmatrix} 1 \\ -5 \\ -1 \end{pmatrix} = \begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix}.\)