question_answer

Let \[\vec{a}=\hat{i}+\hat{j}+\sqrt{2}\,\hat{k},\] \[\vec{b}={{b}_{1}}\,\hat{i}+{{b}_{2}}\hat{j}+\sqrt{2}\hat{k}\] and \[\vec{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k}\] 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 \[\left| {\vec{b}} \right|\] is equal to

Answer:

6