$ABCDEF$ is a regular hexagon whose centre is $O$. Then, $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF}$ is equal to (in $\vec{AO}$)

Let the vectors $\overrightarrow{AB} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\overrightarrow{AC} = 2\hat{i} + 4\hat{j} + 4\hat{k}$ be two sides of a triangle $ABC$. If $G$ is the centroid of $\triangle ABC$, then $\frac{27}{7}(\overrightarrow{AG})^2 + 5 =$

Let $\bar{a}$ and $\bar{b}$ be the position vectors of points $A$ and $B$ respectively. $C$ and $D$ are points on the line $AB$ such that $\overline{AC} = 3 \overline{AB}$ and $\overline{BD} = 2 \overline{BA}$. Find the vector $\overline{CD}$.

If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors,then which of the following is true?

If the vectors $3\hat{i} - 2\hat{j} + 5\hat{k}$ and $-2\hat{i} + p\hat{j} - q\hat{k}$ are parallel,then find the values of $p$ and $q$.

If $P$ divides the line segment joining the points $A$ and $B$ in the ratio $2:1$ and the position vectors of $A$ and $B$ are $\hat{i}-2\hat{j}$ and $-3\hat{i}+5\hat{j}$ respectively,then the position vector of $P$ is