The position vectors of points $P$ and $Q$ are $\vec{p}$ and $\vec{q}$ respectively. The point $R$ divides line segment $PQ$ in the ratio $3:1$ and $S$ is the mid-point of line segment $PR$. The position vector of $S$ is:

$\frac{5\vec{p} + 3\vec{q}}{8}$

The correct answer is Option (4) → $\frac{5\vec{p} + 3\vec{q}}{8}$ ##

Position vector of $R$,

$\vec{r} = \frac{3\vec{q} + \vec{p}}{3+1}$

$\text{or} \quad \vec{r} = \frac{3\vec{q} + \vec{p}}{4}$

Given $S$ is the mid-point of $PR$

$∴$ Position vector of $S$,

$\vec{s} = \frac{\vec{p} + \vec{r}}{2}$

$\vec{s} = \frac{\vec{p} + \left( \frac{3\vec{q} + \vec{p}}{4} \right)}{2}$

$\vec{s} = \frac{5\vec{p} + 3\vec{q}}{8}$