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Give explanation of position and displacement vectors for particle moving in a plane by giving suitable equations.
Solution
Position vector : The position vector $\vec{r}$ of a particle P located in a plane with reference to the origin,
$\vec{r}=x \hat{i}+y \hat{j}$
Where $x$ and $y$ are components of $\vec{r}$ along $x$ and $y$-axes or simply they are the coordinates of the object.
Displacement vector :
$(b)$
Suppose a particle moves along the curve shown by the thick line and is at $\mathrm{P}$ at time $t$ and $\mathrm{P}^{\prime}$ at time $t^{\prime}$ at $\mathrm{P}, \vec{r}=x \hat{i}+y \hat{j}$ at $\mathrm{P}^{\prime}, \overrightarrow{r^{\prime}}=x^{\prime} \hat{i}+y^{\prime} \hat{j}$
Then, the displacement is $\overrightarrow{r_{1}}=x_{1} \hat{i}+y_{1} \hat{j}$ and is directed from $\mathrm{P}$ to $\mathrm{P}^{\prime}$.
$\overrightarrow{\Delta r} &=\left(x^{\prime}-x\right) \hat{i}+\left(y^{\prime}-y\right) \hat{j}$
$=\Delta x \hat{i}+\Delta y \hat{j}$
where $\Delta x=x^{\prime}-x, \Delta y=y^{\prime}-y$
