The line \( L_{1} \) is parallel to the vector \( \vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k} \) and passes through the point \( (7, 6, 2) \) and the line \( L_{2} \) is parallel to the vector \( \vec{b} = 2\hat{i} + \hat{j} + 3\hat{k} \) and passes through the point \( (5, 3, 4) \). The shortest distance between the lines \( L_{1} \) and \( L_{2} \) is:
