## Displacement

The **displacement** of a body in a time interval is equivalent to the change of its position in that interval. Since the position of a body is a vector quantity, the displacement of a body is also a vector.

**displacement**, of a body between the positions

*Pi*and

*Pf*is defined as the vector difference of the position vectors of the defined points

*Pi*and

*Pf*. Its expression, in Cartesian coordinates, is given by:

Where:

- $\u2206\overrightarrow{r}$ :
**Displacement vector**or**displacement** - ${\overrightarrow{r}}_{i}$, ${\overrightarrow{r}}_{f}$ : Position vectors of the points where the body is located at the beginning (
*Pi*) and the end (*Pf*) of the movement - x
_{i}, x_{f},y_{i}, y_{f}, z_{i}, z_{f}: coordinates*x*,*y*and*z*of the points*Pi*and*Pf*

The **unit of measurement** of displacement is the **meter [m]** and in three dimensions its magnitude is given by the following expression:

It is important to realize that *a body may move between two instants of time and yet its displacement may be 0*. This always happens when the initial and final positions of the body, in the time range studied, are the same. In the following picture, you can see the displacement vector and the various concepts presented in a three-dimensional space.

In the case where we are analyzing the problem in fewer dimensions, we can eliminate the unnecessary coordinates, simplifying the above expressions. The displacement vector expression would become:

- In two dimensions $\u2206\overrightarrow{r}={\overrightarrow{r}}_{f}-{\overrightarrow{r}}_{i}=\left({x}_{f}-{x}_{i}\right)\overrightarrow{i}+\left({y}_{f}-{y}_{i}\right)\overrightarrow{j}+\overline{)({z}_{f}-{z}_{i})\overrightarrow{k}}=\left({x}_{f}-{x}_{i}\right)\overrightarrow{i}+\left({y}_{f}-{y}_{i}\right)\overrightarrow{j}$, and its magnitude would be given by the expression $\left|\u2206\overrightarrow{r}\right|=\sqrt{{\left({x}_{f}-{x}_{i}\right)}^{2}+{\left({y}_{f}-{y}_{i}\right)}^{2}+\overline{){\left({z}_{f}-{z}_{i}\right)}^{2}}}=\sqrt{{\left({x}_{f}-{x}_{i}\right)}^{2}+{\left({y}_{f}-{y}_{i}\right)}^{2}}$, since
*(z*_{f}-z_{i})=0 - In one dimension $\u2206\overrightarrow{r}={\overrightarrow{r}}_{f}-{\overrightarrow{r}}_{i}=\left({x}_{f}-{x}_{i}\right)\overrightarrow{i}+\overline{)\left({y}_{f}-{y}_{i}\right)\overrightarrow{j}}+\overline{)({z}_{f}-{z}_{i})\overrightarrow{k}}=\left({x}_{f}-{x}_{i}\right)\overrightarrow{i}$, and its magnitude would be given by the expression $\left|\u2206\overrightarrow{r}\right|=\sqrt{{\left({x}_{f}-{x}_{i}\right)}^{2}+\overline{){\left({y}_{f}-{y}_{i}\right)}^{2}}+\overline{){\left({z}_{f}-{z}_{i}\right)}^{2}}}={x}_{f}-{x}_{i}$, since
*(y*and_{f}-y_{i})=0*(z*_{f}-z_{i})=0