Explain position and displacement vectors. How the magnitude of vector quantity is represented ?

Position vector: To describe the position of an object moving in a plane, we need to choose a convenient point, say $\mathrm{O}$ as origin.

Let $\mathrm{P}$ and $\mathrm{P}^{\prime}$ be the positions of the object at time $t$ and $t^{\prime}$, respectively from figure $(a)$. $\overrightarrow{\mathrm{OP}}$ is the position vector of the object at time $t$. It is represented by a symbol $\vec{r}$.

Point $\mathrm{P}^{\prime}$ is represented by another position vector. $\overrightarrow{\mathrm{OP}^{\prime}}$ denoted by $\overrightarrow{r^{\prime}}$.

The length of the vector $\vec{r}$ represents the magnitude of the vector and its direction is the direction in which $P$ lies as seen from $O$.

Displacement vector : If the object moves from $\mathrm{P}$ to $\mathrm{P}^{\prime}$, the vector $\overrightarrow{P P}^{\prime}$ (with tail at $\mathrm{P}$ and tip at P') is called the displacement vector corresponding to motion from point $P$ (at time $t$ ) to point $P'$ (at time $t^{\prime}$ ).

Important note :

$(1)$ Displacement vector is the straight line joining the initial and final position.

$(2)$ It does not depend on the actual path undertaken by the object between the two positions. For example, in figure $(b)$ given the initial and final positions as $\mathrm{P}$ and $\mathrm{Q}$, the displacement

vector is the same $\overrightarrow{P Q}$ for different paths of journey, say $PABCQ$, $PDQ$ and $PBEFQ.$

$(3)$ Therefore, the magnitude of displacement is either less or equal to the path length of an object between two points.