4 The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = \left( \begin{array} { c } p - 1
q + 2
2 r - 3 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { c } 2 p + 4
2 q - 5
r + 3 \end{array} \right)\), where \(p , q\) and \(r\) are real numbers.
- Given that \(\mathbf { b }\) is not a multiple of \(\mathbf { a }\) and that \(\mathbf { a } \times \mathbf { b } = \mathbf { 0 }\), determine all possible sets of values of \(p , q\) and \(r\).
- You are given instead that \(\mathbf { b } = \lambda \mathbf { a }\), where \(\lambda\) is an integer with \(| \lambda | > 1\).
By writing each of \(p , q\) and \(r\) in terms of \(\lambda\), show that there is a unique value of \(\lambda\) for which \(p , q\) and \(r\) are all integers, stating this set of values of \(p , q\) and \(r\).