question_answer

Let \[\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\,\,\,\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}\] and \[\,\,\overrightarrow{c}=\hat{i}+\hat{j}-2\hat{k}\] be three vectors. A vector of the type \[\overrightarrow{b}+\lambda \overrightarrow{c}\] for some scalar \[\lambda \], whose projection on \[\overrightarrow{a}\] is of magnitude \[\sqrt{\frac{2}{3}}\] is

A)  \[2\hat{i}+3\hat{j}+3\hat{k}\]  

B)  \[2\hat{i}+\hat{j}+5\hat{k}\]

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

D)  \[2\hat{i}-\hat{j}+5\hat{k}\]