If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}, \overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{d}=\hat{i}-\hat{j}-\hat{k}$, then match the following List-$I$ with List-$II$:
List-$I$ List-$II$ $(i)$ $\overrightarrow{a} \cdot \overrightarrow{b}$ $(A)$ $\overrightarrow{a} \cdot \overrightarrow{d}$ (ii) $\overrightarrow{b} \cdot \overrightarrow{c}$ $(B)$ $3$ (iii) $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ $(C)$ $\overrightarrow{b} \cdot \overrightarrow{d}$ (iv) $\overrightarrow{b} \times \overrightarrow{c}$ $(D)$ $2\hat{i}-2\hat{k}$ $(E)$ $2\hat{j}+2\hat{k}$ $(F)$ $4$
| List-$I$ | List-$II$ |
| $(i)$ $\overrightarrow{a} \cdot \overrightarrow{b}$ | $(A)$ $\overrightarrow{a} \cdot \overrightarrow{d}$ |
| (ii) $\overrightarrow{b} \cdot \overrightarrow{c}$ | $(B)$ $3$ |
| (iii) $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ | $(C)$ $\overrightarrow{b} \cdot \overrightarrow{d}$ |
| (iv) $\overrightarrow{b} \times \overrightarrow{c}$ | $(D)$ $2\hat{i}-2\hat{k}$ |
| $(E)$ $2\hat{j}+2\hat{k}$ | |
| $(F)$ $4$ |
- A$i-C, ii-A, iii-B, iv-F$
- B$i-C, ii-A, iii-F, iv-E$
- C$i-A, ii-C, iii-B, iv-F$
- D$i-A, ii-C, iii-F, iv-D$
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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 $\vec{u}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$,$-t \alpha+\beta+\gamma=0$,$\alpha-t \beta+\gamma=0$,and $\alpha+\beta-t \gamma=0$. Match each entry in List-$I$ to the correct entry in List-$II$ and choose the correct option.
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| List-$I$ | List-$II$ |
| $(P)$ $|\vec{v}|^2$ 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 | $(3)$ $2$ |
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| $(5)$ $5$ |
If $\vec{a}+l \vec{b}+l^2 \vec{c}=0$ and $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=3(\vec{b} \times \vec{c})$, then the minimum value of such $l$ is
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