In a quadrilateral $PQRS$,$M$ and $N$ are mid-points of the sides $PQ$ and $RS$ respectively. If $\vec{PS} + \vec{QR} = t \vec{MN}$,then $t =$

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors of $A, B, C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac{\vec{a} + \vec{b} + \vec{c}}{4}$ respectively,then the position vector of the orthocentre of this triangle is:

Classify the following measure as a scalar or a vector: $20 \, m/s^{2}$.

If $a, b, c$ are three non-coplanar vectors such that $a + b + c = \alpha d$ and $b + c + d = \beta a$,then $a + b + c + d$ is equal to

Let $O$ be the origin and the position vectors of $A$ and $B$ be $2 \hat{i}+2 \hat{j}+\hat{k}$ and $2 \hat{i}+4 \hat{j}+4 \hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$,then the length of $OC$ is

If the position vector of a point $A$ is $a + 2b$ and a point $P$ divides $AB$ in the ratio $2:3$,where the position vector of $P$ is $a$,then the position vector of $B$ is: