Vector Algebra

If \(\bar{a}=\hat{\boldsymbol{i}}+\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}, \bar{b}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}\) and \(\bar{c}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}-\hat{\boldsymbol{k}}\) are three vectors then vector \(\bar{r}\) in the plane of \(\bar{a}\) and \(\bar{b}\), whose projection on \(\bar{c}\) is \(\frac{1}{\sqrt{3}}\), is given by