If $G$ and $G'$ are the centroids of the triangles $ABC$ and $A'B'C'$ respectively,then $\overrightarrow{AA'} + \overrightarrow{BB'} + \overrightarrow{CC'} = $

If the position vectors of points $P, Q, R,$ and $S$ are $2\hat{i} + 3\hat{j} + 5\hat{k}$,$\hat{i} + 2\hat{j} + 3\hat{k}$,$-5\hat{i} + 4\hat{j} - 2\hat{k}$,and $\hat{i} + 10\hat{j} + 10\hat{k}$ respectively,then:

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

In a right-angled triangle,if the position vector of the vertex having the right angle is $\vec{A} = -3\hat{i} + 5\hat{j} + 2\hat{k}$ and the position vector of the midpoint of its hypotenuse is $\vec{M} = 6\hat{i} + 2\hat{j} + 5\hat{k}$,then the position vector of its centroid is

$\bar{a}$ and $\bar{b}$ are non-collinear vectors. If $\bar{p} = (2x + 1)\bar{a} - \bar{b}$ and $\bar{q} = (x - 2)\bar{a} + \bar{b}$ are collinear vectors,then $x =$

Let $ABC$ be a triangle and $\bar{a}, \bar{b}, \bar{c}$ be the position vectors of $A, B, C$ respectively. Let $D$ divide $BC$ in the ratio $3:1$ internally and $E$ divide $AD$ in the ratio $4:1$ internally. Let $BE$ meet $AC$ in $F$. If $E$ divides $BF$ in the ratio $3:2$ internally, then the position vector of $F$ is