RI Vectors Tutorial Q9

The points \({P}\) and \({Q}\) have position vectors \({\mathbf{a} + \mathbf{b}}\) and \({3\mathbf{a} - 2\mathbf{b}}\) respectively relative to the origin \({O}\).

Given that \({OPQR}\) is a parallelogram, express the vectors \({\overrightarrow{PQ}}\) and \({\overrightarrow{PR}}\) in terms of \({\mathbf{a}}\) and \({\mathbf{b}}\).

By evaluating two scalar products, show that if \({OPQR}\) is a square, then \({\left| \mathbf{a} \right|^2 = 2\left| \mathbf{b} \right|^2}\).