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.
| List-I | List-II |
|---|---|
| (P) The value of $$\gamma^2 + \delta^2$$ is | (1) 0 |
| (Q) If $$x\vec{u} + y\vec{v} + z\vec{w} = \hat{j}$$ for some real numbers $x$, $y$ and $z$, then the value of $x$ is | (2) 1 |
| (R) The value of $$\left|\vec{u} \cdot (\vec{v} \times \vec{w})\right|$$ is | (3) $$\frac{1}{\sqrt{2}}$$ |
| (S) The value of $$\left|\vec{u} \times (\vec{v} \times \vec{w})\right|$$ is | (4) $$\frac{1}{\sqrt{3}}$$ |
| (5) $$\frac{5}{6}$$ |
JEE Advanced Subjects
Browse all chapters by subject