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.

List – I	List – II
(P) $\lvert \vec{v} \rvert^2$ is equal to	(1) 0
(Q) If $\alpha = \sqrt{3}$, then $\gamma^2$ is equal to	(2) 1
(R) If $\alpha = \sqrt{3}$, then $(\beta + \gamma)^2$ is equal to	(3) 2
(S) If $\alpha = \sqrt{2}$, then $t + 3$ is equal to	(4) 3
	(5) 5

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$, $\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$, respectively. Then which of the following statements is true?

Let O be the origin and let PQR be an arbitrary triangle. The point S is such that

$$\overrightarrow{OP}$$ . $$\overrightarrow{OQ}$$ + $$\overrightarrow{OR}$$ . $$\overrightarrow{OS}$$ = $$\overrightarrow{OR}$$ . $$\overrightarrow{OP}$$ + $$\overrightarrow{OQ}$$ . $$\overrightarrow{OS}$$ = $$\overrightarrow{OQ}$$ . $$\overrightarrow{OR}$$ + $$\overrightarrow{OP}$$ . $$\overrightarrow{OS}$$

Then the triangle PQR has S as its

Let O be the origin and $$\overrightarrow{OX}$$, $$\overrightarrow{OY}$$, $$\overrightarrow{OZ}$$ be three unit vectors in the directions of the sides $$\overrightarrow{QR}$$, $$\overrightarrow{RP}$$, $$\overrightarrow{PQ}$$ respectively, of a triangle PQR.

|$$\overrightarrow{OX}$$ $$ \times $$ $$\overrightarrow{OY}$$| = ?