Let $ \vec{a}, \vec{b} $ be two vectors, and let P, Q and R be the points with position vectors $ \vec{a}, \vec{b} $ and $ \vec{a} + \vec{b} $, respectively, with respect to the origin O. If $ |\vec{a} + \vec{b}| = \sqrt{21} $, $ |\vec{a} - \vec{b}| = 3 $, and $ \vec{a} $ and $ (\vec{a} - \vec{b}) $ are perpendicular to each other, then the area of the triangle OPR is :

For real numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and $\mu$, consider the matrix

$$ M = \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{bmatrix}. $$

Suppose that $MM^{T} = I$, where $M^{T}$ is the transpose of the matrix $M$, and $I$ is the $3 \times 3$ identity matrix. Let

$$ \vec{u} = \alpha\,\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\,\hat{k}, \qquad \vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k} \qquad \text{and} \qquad \vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}. $$

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{v} \times \vec{w} = \vec{u}$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that

$\vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k},\ \ \ - t \alpha + \beta + \gamma = 0,\ \ \ \alpha - t \beta + \gamma = 0,\ \ \ \alpha + \beta - t \gamma = 0.$

Match each entry in List-I to the correct entry in List-II and choose the correct option.