Relative to an origin \(O\), the position vectors of two points \(P\) and \(Q\) are \(\mathbf { p }\) and \(\mathbf { q }\) respectively. The point \(R\) is such that \(P Q R\) is a straight line with \(Q\) the mid-point of \(P R\). Find the position vector of \(R\) in terms of \(\mathbf { p }\) and \(\mathbf { q }\), simplifying your answer.
The vector \(6 \mathbf { i } + a \mathbf { j } + b \mathbf { k }\) has magnitude 21 and is perpendicular to \(3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\). Find the possible values of \(a\) and \(b\), showing all necessary working.