Three vectors $\vec{P}$, $\vec{Q}$ and $\vec{R}$ are shown in the figure. Let $S$ be any point on the vector $\vec{R}$. The distance between the points $P$ and $S$ is $b|\vec{R}|$. The general relation among vectors $\vec{P}$, $\vec{Q}$ and $\vec{S}$ is

Step 1: Define the vector $\vec{R}$.

The vector $\vec{R}$ connects point $P$ to point $Q$.

$$\vec{R} = \vec{Q} - \vec{P}$$

Step 2: Express the vector $\overrightarrow{PS}$.

Since $S$ lies on the vector $\vec{R}$ at a distance of $b|\vec{R}|$ from $P$, the vector $\overrightarrow{PS}$ is a fraction $b$ of $\vec{R}$.

$$\overrightarrow{PS} = b\vec{R}$$

Step 3: Express the position vector $\vec{S}$.

By vector addition, the position vector $\vec{S}$ is the sum of the position vector $\vec{P}$ and the displacement vector $\overrightarrow{PS}$.

$$\vec{S} = \vec{P} + \overrightarrow{PS}$$ $$\vec{S} = \vec{P} + b\vec{R}$$

Step 4: Substitute $\vec{R}$ to obtain the relation.

Substitute $\vec{R} = \vec{Q} - \vec{P}$ into the equation for $\vec{S}$.

$$\vec{S} = \vec{P} + b(\vec{Q} - \vec{P})$$ $$\vec{S} = (1 - b)\vec{P} + b\vec{Q}$$

Answer:

$$\vec{S} = (1 - b)\vec{P} + b\vec{Q}$$