$OABCD$ is a pentagon in which the sides $OA$ and $CB$ are parallel and the sides $OD$ and $AB$ are parallel. Also,it is given that $\frac{OA}{CB}=2$,$\frac{OD}{AB}=\frac{1}{3}$. If $\vec{OA}=\vec{a}, \vec{OD}=\vec{d}$,then $\vec{AD}+\vec{OC}+\vec{DC}=$

Find a vector in the direction of vector $\vec{a} = \hat{i} - 2\hat{j}$ that has magnitude $7$ units.

If $\hat{i}-2 \hat{j}+3 \hat{k}$,$2 \hat{i}+3 \hat{j}+\hat{k}$,and $-3 \hat{i}-\hat{j}-2 \hat{k}$ are the position vectors of three points $A$,$B$,and $C$ respectively,then $A$,$B$,and $C$

Suppose that $\bar{p}, \bar{q}$ and $\bar{r}$ are three non-coplanar vectors in $\mathbb{R}^3$. Let the components of a vector $\bar{s}$ along $\bar{p}, \bar{q}$ and $\bar{r}$ be $4, 3$ and $5$ respectively. If the components of this vector $\bar{s}$ along $(-\bar{p}+\bar{q}+\bar{r}), (\bar{p}-\bar{q}+\bar{r})$ and $(-\bar{p}-\bar{q}+\bar{r})$ are $x, y$ and $z$ respectively,then the value of $2x+y+z$ is

If $\vec{AB} = 3 \hat{i} + 5 \hat{j} + 4 \hat{k}$ and $\vec{AC} = 5 \hat{i} - 5 \hat{j} + 2 \hat{k}$ represent the sides of triangle $ABC$,then the length of the median through $A$ is

If $PQ + QR = (2\lambda^2 - 5)RP$,then $\lambda$ is equal to