# Advantages of Polar Coordinate System & Rotating Unit Vectors

• torito_verdejo
In summary, the advantage of using a polar coordinate system with rotating unit vectors is that it is simpler to solve certain problems using these coordinates.
torito_verdejo
What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors ##\mathbf{ \hat{r}}## and ##\mathbf{\hat{\theta}}## have a variable direction, such that for a Cartesian coordinates system's base vectors ##\mathbf{ \hat{i}}## and ##\mathbf{ \hat{j}}## we have
$$\mathbf{\hat{r}} = \cos \theta\ \mathbf{\hat{i}} + \sin \theta\ \mathbf{\hat{j}}$$
$$\mathbf{\hat{\theta}} = -\sin \theta\ \mathbf{\hat{i}} + \cos \theta\ \mathbf{\hat{j}}$$
Now, isn't counter-productive to define a coordinate system in terms of another? Why, at least in this book, we choose to use such a dependent coordinate system, instead of using a polar coordinate system employing a radius and the angle that this one forms with a polar axis, that are therefore independent of another coordinate system?

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torito_verdejo said:
Summary: What is the advantage of using a polar coordinate system with rotating, not constant unit vectors?

Now, isn't counter-productive to define a coordinate system in terms of another?
Things like coordinate systems are usually chosen to make things simpler.

Simpler or not simpler is by definition a matter of opinion, not fact.

anorlunda said:
Things like coordinate systems are usually chosen to make things simpler.

Simpler or not simpler is by definition a matter of opinion, not fact.
Well, people can generally agree on a "simpler" way of doing or modeling things, and in that sense my question is not merely about individual opinion.

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torito_verdejo said:
Well, complexity and hence simplicity are pretty non-subjective, and when it comes to the way we model something, people can generally agree on a "simpler" way of doing things. In any case, my question is clearly answerable: what are the advantages of defining a polar coordinate system that relies on the Cartesian coordinate system? How and why is such a system more appropriate than another polar coordinate system that only makes use of a radius, a polar axis and the angle both of these form?
The polar coordinate system does not "rely" on the Cartesian system. The equations you gave merely shows the relationship between the unit vectors in polar coordinates and the unit vectors in Cartesian coordinates. The polar coordinate unit vectors are not constant, but change direction as a function of the polar angle ##\theta##.

As to the question about the value of using polar coordinates rather than Cartesian coordinates, this comes into play when we find that it is much simpler to solve certain problems using polar coordinates. This applies strongly when solving distributed parameter problems with boundary conditions applied on cylindrical surfaces.

Cryo and torito_verdejo
Chestermiller said:
The polar coordinate system does not "rely" on the Cartesian system. The equations you gave merely shows the relationship between the unit vectors in polar coordinates and the unit vectors in Cartesian coordinates. The polar coordinate unit vectors are not constant, but change direction as a function of the polar angle ##\theta##.

As to the question about the value of using polar coordinates rather than Cartesian coordinates, this comes into play when we find that it is much simpler to solve certain problems using polar coordinates. This applies strongly when solving distributed parameter problems with boundary conditions applied on cylindrical surfaces.
Just to clarify, I was not asking about the advantages of a polar coordinate system over a Cartesian one, but about the advantages of a polar coordinate system based on rotating base vectors over another polar coordinate system with simply a radius, a polar axis and the angle between both. However, you made me realize I was getting confused by this "translation" to the Cartesian coordinate system. Thank you. :)

torito_verdejo said:
Thank you for your answer. Just to clarify, I was not asking about the advantage of a polar coordinate system over a Cartesian one, but about the advantage of a polar coordinate system based on rotating base vectors over another polar coordinate system with simply a radius, a polar axis and the angle between both.
Consider an object moving in a circle under the influence of a radially directed force: a rock on a string, or a planet in a circular orbit, or a bicyclist on a banked circular track. In standard polar coordinates its velocity vector is ##\frac{V}{R}\frac{d\theta}{dt}\hat{\theta}## with ##V/R## a constant.

What does this vector look like using the coordinates that you suggest?

Chestermiller and torito_verdejo
Nugatory said:
Consider an object moving in a circle under the influence of a radially directed force: a rock on a string, or a planet in a circular orbit, or a bicyclist on a banked circular track. In standard polar coordinates its velocity vector is ##\frac{V}{R}\frac{d\theta}{dt}\hat{\theta}## with ##V/R## a constant.

What does this vector look like using the coordinates that you suggest?
Hit and sunk. You just made me understand how useful is to be able, through base vectors, to represent vectors as sums of products. Thank you very much. Excuse me for being so noob.

berkeman
torito_verdejo said:
Hit and sunk. You just made me understand how useful is to be able, through base vectors, to represent vectors as sums of products. Thank you very much. Excuse me for being so noob.
You don't need to excuse yourself for anything. We all went through the same issues, so now we're here to help you. Welcome to Physics Forums.

vanhees71 and torito_verdejo

## 1. What is a polar coordinate system?

A polar coordinate system is a two-dimensional coordinate system used to represent points in a plane. It uses a distance from the origin and an angle from a fixed reference direction to specify the location of a point.

## 2. What are the advantages of using a polar coordinate system?

One advantage of using a polar coordinate system is that it simplifies the representation of complex shapes, such as circles and spirals. It also allows for easier visualization of rotational and symmetrical patterns.

## 3. How are unit vectors used in a polar coordinate system?

In a polar coordinate system, unit vectors are used to represent the direction of the coordinate axes. The unit vector in the radial direction is denoted as ˆr and the unit vector in the angular direction is denoted as ˆθ.

## 4. What is the significance of rotating unit vectors in a polar coordinate system?

Rotating unit vectors in a polar coordinate system can be used to represent the change in direction of a point. This is particularly useful in physics and engineering applications, where the direction of a moving object may change over time.

## 5. Can a polar coordinate system be used in three-dimensional space?

Yes, a polar coordinate system can be extended to three-dimensional space, where it is known as a spherical coordinate system. In this system, a third coordinate, known as the azimuth angle, is used in addition to the distance from the origin and the angle from a reference direction.

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