Let \(\vec{a} = \hat{i} + 2\hat{j} + \hat{k}, \quad \vec{b} = \hat{i} - \hat{j} + \hat{k} \), and \(\vec{c} = \hat{i} + \hat{j} - \hat{k} \). A vector in the plane of \(\vec{a}\) and \(\vec{b}\) whose projection on\( \vec{c}\) is $\frac{1}{\sqrt{3}}$ is ...