Let $\vec{a}=\hat{i}+\hat{j}+\sqrt{2}\hat{k},\vec{b}={\text{b}}_1\hat{i}+b_{\text{2}}\hat{j}+\sqrt{2}\hat{k}$ and $\vec{c}=5\hat{i}+\hat{j}+\sqrt{\text{2}}\hat{k}$ 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 :