Let \( \vec{a}=2\hat{i}-\hat{j}-\hat{k},\ \vec{b}=\hat{i}+3\hat{j}-\hat{k} \) and \( \vec{c}=2\hat{i}+\hat{j}+3\hat{k} \). Let \( \vec{v} \) be the vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \), such that the length of its projection on the vector \( \vec{c} \) is \( \frac{1}{\sqrt{14}} \). Then \( |\vec{v}| \) is equal to:
