Let $\vec{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}},\, \vec{b} = b_1\hat{\mathrm{i}} + b_2\hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ and $\vec{c} = 5\hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $|\vec{a}|$. If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $|\vec{b}|$ is equal to:\\
(1) $\sqrt{22}$\\
(2) $\sqrt{32}$\\
(3) 6\\
(4) 4