Let $\vec{w}=\overset{^}{ı}+\overset{^}{ȷ}-2\overset{^}{k}$, 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 $\vec{u}=\alpha \overset{^}{ı}+\beta \overset{^}{ȷ}+\gamma \overset{^}{k},-t\alpha +\beta +\gamma =0,\alpha -t\beta +\gamma =0,$ $\text{ and }\alpha +\beta -t\gamma =0$ (P) $\ |\vec{v}|^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 Match each entry in List-I to the correct entry in List-II and choose the correct option.

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Let \(\vec{w}=\overset{^}{ı}+\overset{^}{ȷ}-2\overset{^}{k}\), and \(\vec{u}\) and \(\vec{v}\) be two vectors, such that…

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(P)	\(\ \|\vec{v}\|^2 \) 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