Frame of reference and Rectangular Cartesian co-ordinate system

Frame of Reference

The motion of different bodies can be conveniently described if we imagine co-ordinate system attached to a rigid body and the positions of different bodies in space can be described w.r.t. it, then such co-ordinate system is called frame of reference.

For example we at rest relative to earth along with earth and laboratories build on earth etc., constitute a frame of reference. It is with respect to this Earth’s frame of reference that we observe and measure the changes of positions of other objects. Similar frames of reference   can be associated with the objects that may not be at rest relative to us.

Within a frame of reference, we set up co-ordinate system which is used to measure the position of an object and choose our origin accordingly. Here we will assume that the space is flat, three dimensional and Euclidean.

In general three co-ordinate systems are used:

1. Rectangular or cartelism
2. Spherical or polar
3. Cylindrical

In general for the location of an event in a frame of reference or co-ordinate system, we require its position and time of occurrence of the event. So, in this case of rectangular or Cartesian co-ordinate system, four co-ordinates are require to describe the position that is $(x,y,z,t)$. The reference system employed for this purpose is called space time frame of reference.

Rectangular Cartesian co-ordinate system

In mechanics you know the motion of material particle/bodies  when you know its position in space at all instants of time.

In this system three dimensions are represented by three axis x, y, z perpendicular to each other. Co-ordinates of any point in space are taken as distances from origin along the three axes and is written as $(x,y,z)$ This co-ordinate system is generally used when no special symmetry is involved.

Now the the ranges for the coordinate parameters are:

$– \infty \leqslant x \leqslant \infty$

$– \infty \leqslant y \leqslant \infty$

$– \infty \leqslant z \leqslant \infty$

There are two independent rectangular co-ordinate systems possible in space

1. Left handed system
2. Right handed system

These two systems cannot be made to coincide with each other by any means of translation or rotation.

By convention we choose right handed coordinate system is preferred over left handed system and is shown below in the figure

Figure : Cartesian co-ordinate system

Here the position of the particle $P$ relative to the origin $O$ is given by position vector $r$

$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$

Where $\hat{i}$ , $\hat{j}$ and $\hat{k}$ are unit vectors in directions of $X$, $Y$ and $Z$ axes.