The points \(A\) and \(B\) have position vectors \(\overrightarrow{OA}=2i+2j-k\) and \(\overrightarrow{OB}=4i+2j+4k\) relative to the origin, \(O\).

(a) Show that the perpendicular distance from \(A\) to the line through \(O\) and \(B\) is \(\frac13\sqrt{65}\).

(b) The point \(C\) has position vector \(\overrightarrow{OC}=3i+pj+qk\), where \(p\) and \(q\) are constants.

Given that

• \(\overrightarrow{OC}\) is perpendicular to \(\overrightarrow{AB}\) and
• angle \(AOC=\) angle \(COB\),

find the values of \(p\) and \(q\).