3d Geometry

Let \(\overrightarrow a = \widehat i + \widehat j + 2\widehat k\), \(\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j + \widehat k\) be three given vectors. Let \(\overrightarrow v\) be a vector in the plane of \(\overrightarrow a\) and \(\overrightarrow b\) whose projection on \(\overrightarrow c\) is \({2 \over {\sqrt 3 }}\). If \(\overrightarrow v \,.\,\widehat j = 7\), then \(\overrightarrow v \,.\,\left( {\widehat i + \widehat k} \right)\) is equal to :