$ABCD$ is a parallelogram and $P$ is the mid-point of the side $AD$. The line $BP$ meets the diagonal $AC$ in $Q$. Then, the ratio of $AQ:QC$ is equal to

Let $\vec{a}, \vec{b}, \vec{c}$ be co-initial vectors and $\vec{a}=2 \hat{i}-\hat{j}+5 \hat{k}$ and $\vec{b}=3 \hat{i}+7 \hat{j}-\hat{k}$. Let $(\vec{a}, \vec{b})=\theta$ be an acute angle and $\vec{c}$ be the vector along the bisector of the angle $\theta$. If $\lambda, x, y \in R$,then $\vec{c}=$

In a regular hexagon $ABCDEF$,$\overrightarrow{AE} = $

Let $O$ be the origin,$A$ and $B$ be two points with position vectors $-3 \hat{i}-3 \hat{j}+4 \hat{k}$ and $4 \hat{i}-4 \hat{j}-3 \hat{k}$ respectively. Let $P$ be a point such that the line drawn through $P$ parallel to $\overrightarrow{OB}$ meets $OA$ in $L$ and another line through $P$ parallel to $\overrightarrow{OA}$ meets $OB$ in $M$. If $L$ divides $OA$ in the ratio $2:3$ and $M$ divides $OB$ in the ratio $3:2$,then the distance from $O$ to $P$ is

The vectors $\overline{AB} = 3\hat{i} + 4\hat{k}$ and $\overline{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ represent the sides of a triangle $ABC$. What is the length of the median passing through vertex $A$?

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 $\overline{CD}$.