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 $\vec{v}=\vec{v}_1+\overrightarrow{v_2}$, 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

If $\overrightarrow{\mathrm{a}}$ is a nonzero vector such that its projections on the vectors $2 \hat{i}-\hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-2 \hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\overrightarrow{\mathrm{a}}$ is :

Let $ \hat{a} $ be a unit vector perpendicular to the vectors $ \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} $ and $ \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} $, and $ \hat{a} $ makes an angle of $ \cos^{-1} \left( -\frac{1}{3} \right) $ with the vector $ \hat{i} + \hat{j} + \hat{k} $. If $ \hat{a} $ makes an angle of $ \frac{\pi}{3} $ with the vector $ \hat{i} + \alpha\hat{j} + \hat{k} $, then the value of $ a $ is: