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 $\overrightarrow {{v_2}} $ is perpendicular to $\vec u$ . Then the value ${\left| {\overrightarrow {{v_1}} } \right|^2} + {\left| {\overrightarrow {{v_2}} } \right|^2}$ is equal to