Vector Algebra

Let \(\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k\) and \(\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k\). Let \({\overrightarrow \beta _1}\) be parallel to \(\overrightarrow \alpha\) and \({\overrightarrow \beta _2}\) be perpendicular to \(\overrightarrow \alpha\). If \(\overrightarrow \beta = {\overrightarrow \beta _1} + {\overrightarrow \beta _2}\), then the value of \(5{\overrightarrow \beta _2}\,.\left( {\widehat i + \widehat j + \widehat k} \right)\) is :