\(|2\hat a + \hat b + \hat c|\) represents the distance between the points (0, 0, 0) and (2, 1, 1) in the \((\hat a, \hat b, \hat c)\) space.
Thus, by distance formula, we have
\(|2\hat a + \hat b + \hat c| \)
\(= \sqrt {2^2 + 1^2 + 1^2}\)
\(= \sqrt 6\)