Let $\overrightarrow{OA}=2 \hat{i}-3 \hat{j}+\hat{k}$,$\overrightarrow{OB}=\hat{i}-4 \hat{j}-3 \hat{k}$,and $\overrightarrow{OC}=-3 \hat{i}+\hat{j}+2 \hat{k}$ be the position vectors of three points $A$,$B$,and $C$ respectively. If $G$ is the centroid of triangle $ABC$,then $BC^2+CA^2+AB^2+9(OG)^2=$

In a quadrilateral $ABCD$,the point $P$ divides $DC$ in the ratio $1:2$ and $Q$ is the midpoint of $AC$. If $\overrightarrow{AB}+2\overrightarrow{AD}+\overrightarrow{BC}-2\overrightarrow{DC}=k\overrightarrow{PQ}$,then $k$ is equal to

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=2\hat{i}+2\hat{j}+\hat{k}$ and $\vec{d}=\vec{a} \times \vec{b}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c}=|\vec{c}|$,$|\vec{c}-2\vec{a}|^2=8$ and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$,then $|10-3\vec{b} \cdot \vec{c}|+|\vec{d} \times \vec{c}|^2$ is equal to . . . . . .

The minimum value of $(x_1 - x_2)^2 + (\sqrt{2 - x_1^2} - \frac{9}{x_2})^2$ where $x_1 \in (0, \sqrt{2})$ and $x_2 \in R^+$.

Given that $a$ and $b$ are two unit non-collinear vectors,if $u = a - (a \cdot b)b$ and $v = a \times b$,then find $|v| =$.

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 $\frac{\pi}{3}$