Vector Algebra

Let \(\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j - \widehat k\). Let a vector \(\overrightarrow v\) be in the plane containing \(\overrightarrow a\) and \(\overrightarrow b\). If \(\overrightarrow v\) is perpendicular to the vector \(3\widehat i + 2\widehat j - \widehat k\) and its projection on \(\overrightarrow a\) is 19 units, then \({\left| {2\overrightarrow v } \right|^2}\) is equal to _____________.