Interpret the figure and notation

The vectors $\overrightarrow P$ and $\overrightarrow Q$ represent the position vectors of two points $P$ and $Q$ from the origin.

The vector $\overrightarrow R$ is the vector from point $P$ to point $Q$, so $\overrightarrow R = \overrightarrow Q - \overrightarrow P.$

Point $S$ lies somewhere on the vector $\overrightarrow R$, i.e. on the line segment from $P$ to $Q$.

Use the given distance condition

We are told that the distance between points $P$ and $S$ is $PS = b\,|\overrightarrow R|.$

Since $S$ lies on the vector $\overrightarrow R$ starting from $P$, the vector from $P$ to $S$ is a fraction $b$ of $\overrightarrow R$: $\overrightarrow{PS} = b\overrightarrow R.$

Write position vector of $S$

The position vector of $S$ is $\overrightarrow S = \overrightarrow P + \overrightarrow{PS}.$

Substituting $\overrightarrow{PS} = b\overrightarrow R$, $\overrightarrow S = \overrightarrow P + b\overrightarrow R.$

Substitute $\overrightarrow R = \overrightarrow Q - \overrightarrow P$

$\overrightarrow S = \overrightarrow P + b(\overrightarrow Q - \overrightarrow P).$

Expanding, $\overrightarrow S = \overrightarrow P + b\overrightarrow Q - b\overrightarrow P$ $\overrightarrow S = (1-b)\overrightarrow P + b\overrightarrow Q.$

Match with the options

This matches: $\boxed{\overrightarrow S = (1-b)\overrightarrow P + b\overrightarrow Q}$

So the correct option is A.