Vector Algebra

Let \(\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k\) and \(\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k\), where \(\alpha \in R\). If the area of the parallelogram whose adjacent sides are represented by the vectors \(\overrightarrow a\) and \(\overrightarrow b\) is \(\sqrt {15({\alpha ^2} + 4)}\), then the value of \(2{\left| {\overrightarrow a } \right|^2} + \left( {\overrightarrow a \,.\,\overrightarrow b } \right){\left| {\overrightarrow b } \right|^2}\) is equal to :