Let a = i + j +√ 2 k, b = b_1 i + b_2 j +√ 2 k and c =5 i + j +√ 2 k be three vectors such that the projection vector of b on a is a. If a + b is…

Let $\displaystyle \overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\sqrt{2} \hat{k}, \quad \overrightarrow{\mathrm{~b}}=\mathrm{b}_1 \hat{i}+\mathrm{b}_2 \hat{j}+\sqrt{2} \hat{k}$ and $\displaystyle \overrightarrow{\mathrm{c}}=5 \hat{i}+\hat{j}+\sqrt{2} \hat{k}$ be three vectors such that the projection vector of $\displaystyle \vec{b}$ on $\displaystyle \vec{a}$ is $\displaystyle \vec{a}$. If $\displaystyle \vec{a}+\vec{b}$ is perpendicular to $\displaystyle \vec{c}$, then $\displaystyle |\vec{b}|$ is equal to :