Consider two vectors \( \vec{u} = 3\hat{i} - \hat{j} \) and \( \vec{v} = 2\hat{i} + \hat{j} - \lambda \hat{k}, \ \lambda > 0 \). The angle between them is given by \( \cos^{-1}\!\left(\frac{\sqrt{5}}{2\sqrt{7}}\right) \). Let \( \vec{v} = \vec{v}_1 + \vec{v}_2 \), where \( \vec{v}_1 \) is parallel to \( \vec{u} \) and \( \vec{v}_2 \) is perpendicular to \( \vec{u} \). Then the value of \( \lvert \vec{v}_1 \rvert^2 + \lvert \vec{v}_2 \rvert^2 \) is equal to:
