Consider two vectors \(\vec{\mathrm{u}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\vec{\mathrm{v}}=2\hat{\mathrm{i}}+\h…

Consider two vectors \(\vec{\mathrm{u}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\vec{\mathrm{v}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}},\lambda >0\). 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 \(\vec{\mathrm{u}}\) and \({\vec{\mathrm{v}}}_{2}\) is perpendicular to \(\vec{\mathrm{u}}\). Then the value \({\left|{\vec{v}}_{1}\right|}^{2}+{\left|{\vec{v}}_{2}\right|}^{2}\) is equal to

[JEE Main 2025, 4 Apr (Shift 1)]