Let $A$ and $B$ be two points. The position vector of $A$ is $6b - 2a$. Point $P$ divides the line segment $AB$ in the ratio $1 : 2$. If $a - b$ is the position vector of $P$,what is the position vector of $B$?

In the given figure,if a vector $x$ satisfies the equation $x - w = v$,then $x = ?$

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 $x_2 \in R^+$.

Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$,$\vec{b}=\hat{i}+b_2\hat{j}+b_3\hat{k}$ $(b_2, b_3 \in \mathbb{R})$,and $\vec{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ $(c_1, c_2, c_3 \in \mathbb{R})$ be three vectors such that $b_2b_3 > 0$,$\vec{a} \cdot \vec{b} = 0$ and $\begin{bmatrix} 0 & -c_3 & c_2 \\ c_3 & 0 & -c_1 \\ -c_2 & c_1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} 3-c_1 \\ 1-c_2 \\ -1-c_3 \end{bmatrix}$. Then,which of the following is/are $TRUE$?

If $ABCDEF$ is a regular hexagon,and $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} = k \vec{AD}$,then $k = \dots$

If $\bar{a}, \bar{b}, \bar{c}$ are three vectors such that $|\bar{a}|=\sqrt{31}, 4|\bar{b}|=|\bar{c}|=2$ and $2(\bar{a} \times \bar{b})=3(\bar{c} \times \bar{a})$ and if the angle between $\bar{b}$ and $\bar{c}$ is $\frac{2\pi}{3}$,then $\left|\frac{\bar{a} \times \bar{c}}{\bar{a} \cdot \bar{b}}\right|^2=$