Let $PQRS$ be a quadrilateral. If $M$ and $N$ are midpoints of the sides $PQ$ and $RS$ respectively,then $\overrightarrow{PS} + \overrightarrow{QR} =$

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

The number of values of $m \in R$ for which the vectors $\hat{i}+2 \hat{j}+m \hat{k}$ and $\hat{i}+m \hat{j}+2 \hat{k}$ are collinear is

If $D, E$ and $F$ are respectively mid-points of $AB, AC$ and $BC$ in $\triangle ABC$,then $\overrightarrow{BE} + \overrightarrow{AF}$ is equal to

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 $A$?

In $\triangle ABC$,$L, M, N$ are points on $BC, CA, AB$ respectively,dividing them in the ratio $1:2, 2:3, 3:5$. If the point $K$ divides $AB$ in the ratio $5:3$,then find the value of $\left| \frac{\vec{AL} + \vec{BM} + \vec{CN}}{\vec{CK}} \right|$.