A vector $\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$ lies in the plane of the vectors, $\overrightarrow b = \widehat i + \widehat j$ and $\overrightarrow c = \widehat i - \widehat j + 4\widehat k$. If $\overrightarrow a $ bisects the angle between $\overrightarrow b $ and $\overrightarrow c $, then: