Let $ABCDEF$ be a regular hexagon with the vertices $A, B, C, D, E$ and $F$ in counter-clockwise order. Then,the vector $\vec{AB} + \vec{BC}$ is parallel to:

If $\bar{a}, \bar{b}, \bar{c}$ are three non-zero vectors,no two of them are collinear,$\bar{a}+2 \bar{b}$ is collinear with $\bar{c}$,and $\bar{b}+3 \bar{c}$ is collinear with $\bar{a}$,then $\bar{a}+2 \bar{b}$ is equal to:

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let $S$ be any point on the vector $\vec{R}$. The distance between the point $P$ and $S$ is $b|\vec{R}|$. The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is

If the vectors $\vec{a} = \lambda \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = \sqrt{\lambda} \hat{i} + \sqrt{13} \hat{j}$ have the same magnitude,then the value of $\lambda$ is:

The direction of the vectors $(1, 1, 2)$ and $(2, 1, 0)$ is $.......$

In a regular hexagon $ABCDEF$,$\overrightarrow{AB}=\vec{a}$ and $\overrightarrow{BC}=\vec{b}$,then $\overrightarrow{FA}=$