Step-by-Step Solution

Step 1: Find the position vector of point R.

Since R divides PQ in the ratio 3:1, we can use the section formula to find the position vector of R (\(\vec{r}\)):

\(\vec{r} = \frac{3\vec{q} + 1\vec{p}}{3+1} = \frac{3\vec{q} + \vec{p}}{4}\)

Step 2: Find the position vector of point S.

Since S is the midpoint of PR, we can find the position vector of S (\(\vec{s}\)) using the midpoint formula:

\(\vec{s} = \frac{\vec{p} + \vec{r}}{2}\)

Step 3: Substitute the value of \(\vec{r}\) into the equation for \(\vec{s}\).

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

Step 4: Simplify the expression for \(\vec{s}\).

\(\vec{s} = \frac{\frac{4\vec{p} + 3\vec{q} + \vec{p}}{4}}{2} = \frac{5\vec{p} + 3\vec{q}}{8}\)