Let \(\left[ {\vec a} \right] = {\rm{\hat i}} + {\rm{\hat j}} + \sqrt 2 \widehat {{\rm{k\;}}},\;\vec b = {{\rm{b}}_1}{\rm{\hat i}} + {{\rm{b}}_2}{\rm{\hat j}} + \sqrt 2 \widehat {{\rm{k\;}}}{\rm{\;and\;}}\vec c = 5{\rm{\hat i}} + {\rm{\hat j}} + \sqrt 2 \widehat {{\rm{k}}}\) be three vectors such that the projection vector of \(\vec b{\rm{\;on\;}}\vec a{\rm{\;is\;}}\vec a{\rm{.\;If\;}}\vec a + \vec b\)is perpendicular to \(\vec c,{\rm{\;then\;}}\left| {\vec b} \right|\) is equal to:

• A. \(\sqrt {32}\)
• B. 6
• C. \(\sqrt {22}\)
• D. 4