question_answer

If \[\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}\], and \[\overrightarrow{c}=\hat{i}+\hat{j}-2\hat{k}\]then a vector in the plane of  and whose projection on a is of \[\overrightarrow{b}\]and \[\overrightarrow{c}\]whose projection on \[\overrightarrow{a}\] is of magnitude is\[\sqrt{\frac{2}{3}}\]is

A)  \[2\hat{i}\,+\,3\hat{j}-3\hat{k}\]           

B)  \[2\hat{i}\,+\,3\hat{j}+3\hat{k}\]

C)  \[-2\hat{i}\,-\hat{j}+5\hat{k}\]  

D)  \[2\hat{i}\,-\hat{j}+5\hat{k}\]