Let  $\overrightarrow a = \widehat i + \widehat j + \sqrt 2 \widehat k,$   $\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + \sqrt 2 \widehat k$,    $\overrightarrow c = 5\widehat i + \widehat j + \sqrt 2 \widehat k$   be three vectors such that the projection vector of $\overrightarrow b $ on $\overrightarrow a $ is $\overrightarrow a $.
If   $\overrightarrow a + \overrightarrow b $   is perpendicular to $\overrightarrow c $ , then $\left| {\overrightarrow b } \right|$ is equal to :