# Using polar coordinates to find the distance traveled

• ptolema
In summary, the problem is asking to find the distance between the starting and ending points of a tourist's tour through a city. Each stage consists of 3 segments of length 100 feet, separated by right turns of 60°, and a left turn of 60° between stages. It is solved by representing the displacement using complex numbers and drawing the path on a map. After 2010 stages, the distance between the starting and ending points is zero.
ptolema

## Homework Statement

A tourist takes a tour through a city in stages. Each stage consists of 3 segments of length 100 feet, separated by right turns of 60°. Between the last segment of one stage and the first segment of the next stage, the tourist makes a left turn of 60°. At what distance will the tourist be from his initial position after 2010 stages?

## Homework Equations

ε = e(π/3)i = cos(π/3) + i*sin(π/3) corresponds to a 60° turn to the left
ε6 = 1

## The Attempt at a Solution

I really didn't know how to start, so my attempt might not even be relevant. I am also not very familiar with polar coordinates.

I started trying to do this algebraically, by assuming that e-(π/3)i corresponds to a 60° turn to the right. e-(π/3)i*e-(π/3)i*e(π/3)i since the tourist makes 2 right turns then a left in one stage. This equals e-(π/3)i. At this point, I don't know what to do next, or if I should even continue in this direction.

Imagine you are a tourist and draw your path on the map. Draw a straight line that corresponds to 100 feet, then turn to the right at 60°angle and draw the next segment and so on.

ehild

Last edited:
You titled this "find the distance traveled". Do you understand that that is NOT what this question asks you to find? The problem asks you to find the straight line distance between starting and ending points, NOT the total distance traveled.

HallsofIvy said:
You titled this "find the distance traveled". Do you understand that that is NOT what this question asks you to find? The problem asks you to find the straight line distance between starting and ending points, NOT the total distance traveled.

Yes, I understand that now. My problem is that although I can figure out the angle of the tourist after a given number of stages, I'm stuck on how to find the distance between the starting and ending points.
When I do it graphically, I get a path that looks like partial traces of hexagons. The displacement from 1 stage is 200ft, and after 2 stages, the displacement is 200√3ft. I am not sure how I would generalise this for 2010 stages.

Think in displacement vectors. You can represent the displacement also by complex numbers, as you did in the first post. What is the displacement just after the first stage?

ehild

I actually just tried it a different way. I used your suggestion, ehild, of drawing out the path.

[PLAIN]http://www.privateline.com/Cellbasics/sevencellcluster.gif

So the tourist travels in a path that traces the outside of this hexagon cluster. At least, this is how I ended up coming up with a diagram. Working from this assumption, it takes 6 stages to return to the starting point.
From here, it get a bit trivial. 2010/6 = 335, so he returns to his starting point exactly 335 times. After 2010 stages, he is at his starting point, i.e. his distance from the starting point is 0.

Did I get anywhere with this?

Last edited by a moderator:
You have solved the problem, well done! The distance between the end point and starting point of the route is zero. The distance traveled is different :)
I show my picture, with the displacement during the first stage and during the second stage. They differ with the factor e-iπ/3. The sum of 6 stages is zero.

ehild

#### Attachments

• stages.JPG
12.9 KB · Views: 492

## 1. How do polar coordinates help in finding the distance traveled?

Polar coordinates are a way of representing a point in a two-dimensional coordinate system using a distance from the origin and an angle from a reference direction. This is useful for finding the distance traveled because it allows us to easily calculate the distance between two points using the Pythagorean theorem.

## 2. What is the difference between using polar coordinates and Cartesian coordinates to find distance traveled?

The main difference between polar coordinates and Cartesian coordinates is the way they represent points in a two-dimensional system. While Cartesian coordinates use x and y values, polar coordinates use a distance and an angle. This makes it easier to calculate distances and angles in circular or curved paths, making it more suitable for finding distance traveled.

## 3. How is the distance traveled calculated using polar coordinates?

The distance traveled using polar coordinates is calculated by finding the difference between the initial and final polar coordinates and using the Pythagorean theorem to calculate the distance between them. This gives us the total distance traveled along a curved path.

## 4. Can polar coordinates be used to find the distance traveled in a straight line?

Yes, polar coordinates can also be used to find the distance traveled in a straight line. In this case, the angle between the initial and final point will be 0 degrees, making the calculation simpler as it reduces to just calculating the difference in distance.

## 5. Are polar coordinates always more accurate than Cartesian coordinates when finding distance traveled?

There is no inherent difference in accuracy between polar and Cartesian coordinates when finding distance traveled. However, polar coordinates may be more suitable for certain types of paths, such as circular or curved paths, while Cartesian coordinates may be more suitable for straight line paths. It ultimately depends on the specific situation and the method that is most appropriate for it.

Replies
24
Views
2K
Replies
7
Views
2K
Replies
12
Views
2K
Replies
7
Views
2K
Replies
9
Views
1K
Replies
3
Views
2K
Replies
3
Views
1K
Replies
5
Views
2K
Replies
4
Views
1K
Replies
11
Views
2K