If $D, E, F$ are the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$ respectively,find the position vector of the centroid of $\Delta DEF$,given that the position vectors of $A, B, C$ are $i + j, j + k, k + i$ respectively.

The position vectors of $P$ and $Q$ are respectively $\overrightarrow{a}$ and $\overrightarrow{b}$. If $R$ is a point on the line $PQ$ such that $\overrightarrow{PR}=5 \overrightarrow{PQ}$,then the position vector of $R$ is

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

If $a$ and $b$ represent two non-collinear vectors, the equation $r = ta + (1-t)b$ represents

$A$ vector in the direction of $v = 2\hat{i} + 3\hat{j} + \hat{k}$ with magnitude $\sqrt{7}$ is

In $\triangle PQR$,$(4 \hat{i}+3 \hat{j}+6 \hat{k})$,$(2 \hat{i}+2 \hat{j}+3 \hat{k})$ and $(3 \hat{i}+\hat{j}+3 \hat{k})$ are the position vectors of the vertices $P, Q$ and $R$ respectively. Then the position vector of the point of intersection of the angle bisector of $P$ with $QR$ is