POSITION VECTOR
It is a vector which denotes the position of a particle at any instant of time, with respect to some reference frame or coordinate system.
The position vector \(\vec{r}\) of the particle at a point \(\mathrm{P}\) is given by
\(\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}\)where \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are components of \(\vec{r}\) , Figure 2.25 shows the position vector \(\vec{r}\) .
Figure 2.25 Position vector in Cartesian coordinate system
EXAMPLE 2.13
Determine the position vectors for the following particles which are located at points \(\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}\) .
Solution
The position vector for the point \(\mathrm{P}\) is
\(\vec{r}_{P}=3 \hat{i}\)The position vector for the point \(\mathrm{Q}\) is
\(\vec{r}_{Q}=5 \hat{i}+4 \hat{j}\)The position vector for the point \(\mathrm{R}\) is
\(\vec{r}_{R}=-2 \hat{i}\)The position vector for the point \(S\) is
\(\vec{r}_{s}=3 \hat{i}-6 \hat{j}\)EXAMPLE 2.14
A person initially at rest starts to walk 2 m towards north, then 1 m towards east, then 5 m towards south and then 3 m towards west. What is the position vector of the person at the end of the trip?
Solution
As shown in the Figure, the positive x axis is taken as east direction, positive y direction is taken as north.
After the trip, the person reaches the point $\mathrm{P}$ whose position vector given by
\(\vec{r}=-2 \hat{i}-3 \hat{j}\)The displacement direction is south west.
The displacement direction is south west.