9709 P32 - Mar 2026 - Q11 - 12 marks
4334
The points \(A\) and \(B\) have position vectors \(\overrightarrow{OA}=2\mathbf i+2\mathbf j-\mathbf k\) and \(\overrightarrow{OB}=4\mathbf i+2\mathbf j+4\mathbf k\) relative to the origin, \(O\).
(a) Show that the perpendicular distance from \(A\) to the line through \(O\) and \(B\) is \(\sqrt{65}/3\). [5]
The point \(C\) has position vector \(\overrightarrow{OC}=3\mathbf i+p\mathbf j+q\mathbf k\), where \(p\) and \(q\) are constants.
(b) Given that \(OC\) is perpendicular to \(AB\) and \(\angle AOC=\angle COB\), find the values of \(p\) and \(q\). [7]
