Consider two vectors u =3 i - j and v =2 i + j -λ k, λ>0. The angle between them is given by cos^-1((√ 5)/(2 √ 7)). Let v = v_1 + v_2, where v_1 is…

Consider two vectors $\displaystyle \vec{u}=3 \hat{i}-\hat{j}$ and $\displaystyle \vec{v}=2 \hat{i}+\hat{j}-\lambda \hat{k}, \lambda>0$. The angle between them is given by $\displaystyle \cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)$. Let $\displaystyle \vec{v}=\overrightarrow{v_1}+\overrightarrow{v_2}$, where $\displaystyle \overrightarrow{v_1}$ is parallel to $\displaystyle \vec{u}$ and $\displaystyle \overrightarrow{v_2}$ is perpendicular to $\displaystyle \vec{u}$. Then the value $\displaystyle \left|\overrightarrow{v_1}\right|^2+\left|\overrightarrow{v_2}\right|^2$ is equal to