Vector Algebra

Let \(\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k\) and \(\overrightarrow q = \widehat i + 2\widehat j + \widehat k\) be two vectors. If a vector \(\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \widehat k)\) is perpendicular to each of the vectors (\((\overrightarrow p + \overrightarrow q )\) and \((\overrightarrow p - \overrightarrow q )\), and \(\left| {\overrightarrow r } \right| = \sqrt 3\), then \(\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|\) is equal to _______________.