Vector Algebra

Let \(\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k\), \(\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k\) and \(\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k\) where \(\alpha ,\,\beta \in R\), be three vectors. If the projection of \(\overrightarrow a\) on \(\overrightarrow c\) is \({{10} \over 3}\) and \(\overrightarrow b \times \overrightarrow c = - 6\widehat i + 10\widehat j + 7\widehat k\), then the value of \(\alpha + \beta\) is equal to :