Let $O$ be the origin and $\overline{OA} = 2\hat{i} + 2\hat{j} + \hat{k}$,$\overline{OB} = \hat{i} - 2\hat{j} + 2\hat{k}$ and $\overline{OC} = \frac{1}{2}(\overline{OB} - \lambda\overline{OA})$ for some $\lambda > 0$. If $|\overline{OB} \times \overline{OC}| = \frac{9}{2}$,then which of the following statements is (are) $TRUE$?
$(A)$ Projection of $\overline{OC}$ on $\overline{OA}$ is $-\frac{3}{2}$
$(B)$ Area of the triangle $OAB$ is $\frac{9}{2}$
$(C)$ Area of the triangle $ABC$ is $\frac{9}{2}$
$(D)$ The acute angle between the diagonals of the parallelogram with adjacent sides $\overline{OA}$ and $\overline{OC}$ is $\frac{\pi}{3}$

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ} = a \hat{i} + b \hat{j}$ and $\overrightarrow{PS} = a \hat{i} - b \hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be the projection vectors of $\overrightarrow{w} = \hat{i} + \hat{j}$ along $\overrightarrow{PQ}$ and $\overrightarrow{PS}$,respectively. If $|\vec{u}| + |\vec{v}| = |\vec{w}|$ and if the area of the parallelogram $PQRS$ is $8$,then which of the following statements is/are $TRUE$?
$(A)$ $a + b = 4$
$(B)$ $a - b = 2$
$(C)$ The length of the diagonal $PR$ of the parallelogram $PQRS$ is $4$
$(D)$ $\overrightarrow{w}$ is an angle bisector of the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PS}$

The position vectors of coplanar points $A, B, C,$ and $D$ are $\vec{a}, \vec{b}, \vec{c},$ and $\vec{d}$ respectively,such that $(\vec{a} - \vec{d}) \cdot (\vec{b} - \vec{c}) = 0$ and $(\vec{b} - \vec{d}) \cdot (\vec{c} - \vec{a}) = 0.$ Then the point $D$ of the triangle $ABC$ is

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) $\overrightarrow{b} \times \overrightarrow{c}$	$(D)$ $2\hat{i}-2\hat{k}$
	$(E)$ $2\hat{j}+2\hat{k}$
	$(F)$ $4$

For what values of $\lambda$ are $\vec{a}$ and $\vec{c}$ unit collinear vectors,and given $|\vec{b}| = 6$,if $\vec{b} - 3\vec{c} = \lambda \vec{a}$,then $\lambda = ......$

Let the position vectors of three vertices of a triangle be $4 \overrightarrow{p} + \overrightarrow{q} - 3 \overrightarrow{r}$,$-5 \overrightarrow{p} + \overrightarrow{q} + 2 \overrightarrow{r}$,and $2 \overrightarrow{p} - \overrightarrow{q} + 2 \overrightarrow{r}$. If the position vectors of the orthocenter $(O)$ and the circumcenter $(C)$ of the triangle are $\frac{\overrightarrow{p} + \overrightarrow{q} + \overrightarrow{r}}{4}$ and $\alpha \overrightarrow{p} + \beta \overrightarrow{q} + \gamma \overrightarrow{r}$ respectively,then $\alpha + 2 \beta + 5 \gamma$ is equal to: