If in the given figure $\overrightarrow{OA} = \vec{a}$,$\overrightarrow{OB} = \vec{b}$ and $AP : PB = m : n$,then $\overrightarrow{OP} = $

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors,then what is the maximum value of $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2$?

If $\hat{i}$ is the position vector of the centroid $G$ of triangle $ABC$ and $2\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}-4\hat{k}$ are respectively the position vectors of its vertices $A$ and $B$,then $AG^2+BG^2+CG^2=$

If the magnitudes of $\bar{a}$,$\bar{b}$ and $\bar{a}+\bar{b}$ are respectively $3$,$4$ and $5$,then the magnitude of $\bar{a}-\bar{b}$ is

If $a = i + 2j + 2k$ and $b = 3i + 6j + 2k,$ then a vector in the direction of $a$ and having magnitude as $|b|$ is

If $2 \hat{i}-\hat{j}+\hat{k}$ and $\hat{i}-3 \hat{j}-5 \hat{k}$ are the position vectors of the points $A$ and $B$ respectively,$C$ divides $AB$ in the ratio $2:3$ and $M$ is the mid-point of $AB$,then $5(\text{position vector of } C) - 2(\text{position vector of } M) =$