$A, B, P, Q, R$ are five points in a plane. If the forces acting at point $A$ are $\overline{AP}, \overline{AQ}, \overline{AR}$ and the forces acting at point $B$ are $\overline{PB}, \overline{QB}, \overline{RB}$,find the resultant of all these forces.

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 $\left|\frac{\bar{a} \times \bar{c}}{\bar{a} \cdot \bar{b}}\right|^2=$

If $S$ is the circumcentre,$G$ the centroid,and $O$ the orthocentre of a triangle $ABC$,then $\overrightarrow {SA} + \overrightarrow {SB} + \overrightarrow {SC} = $

Let $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ be two vectors. Consider a vector $\vec{c} = \alpha\vec{a} + \beta\vec{b}$,where $\alpha, \beta \in \mathbb{R}$. If the projection of $\vec{c}$ on the vector $(\vec{a} + \vec{b})$ is $3\sqrt{2}$,then the minimum value of $(\vec{c} - (\vec{a} \times \vec{b})) \cdot \vec{c}$ is equal to:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar vectors and if $\vec{d}$ is such that $\vec{d} = \frac{1}{x}(\vec{a} + \vec{b} + \vec{c})$ and $\vec{d} = \frac{1}{y}(\vec{b} + \vec{c} + \vec{d})$ where $x$ and $y$ are non-zero real numbers, then $\frac{1}{xy}(\vec{a} + \vec{b} + \vec{c} + \vec{d})$ is equal to:

If vectors $\vec{a}$ and $\vec{b}$ represent the sides $\vec{AB}$ and $\vec{BC}$ of a regular hexagon $ABCDEF$,find the vector represented by $\vec{FA}$.