Polar to rectangular coordinates

Click For Summary

Discussion Overview

The discussion revolves around converting a vector expression from polar coordinates to rectangular coordinates, specifically focusing on the expression E = (1/r) ar, where ar is a unit vector. Participants explore the correct representation of the unit vector in rectangular coordinates and the associated mathematical transformations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant attempts to express ar as a simple sum of unit vectors ax and ay, leading to confusion about the correct representation.
  • Another participant points out that adding two unit vectors does not yield another unit vector, highlighting a fundamental misunderstanding in vector decomposition.
  • A participant proposes dividing by the magnitude of the sum of the unit vectors, indicating an attempt to normalize the vector but still arrives at an incorrect expression.
  • Further clarification is provided that ar should be expressed as a weighted sum of ax and ay, specifically ar = (x*ax + y*ay)/r, which is a more accurate representation.
  • Participants express uncertainty about their understanding of vector decomposition and seek confirmation on their approaches.

Areas of Agreement / Disagreement

Participants generally agree that the initial approach to expressing ar was incorrect, but there is no consensus on the correct method until the later clarification is provided. Multiple competing views on vector decomposition remain throughout the discussion.

Contextual Notes

There are limitations in the participants' understanding of vector decomposition, particularly in normalizing vectors and the correct application of polar to rectangular transformations. Some mathematical steps remain unresolved, contributing to ongoing confusion.

Who May Find This Useful

Students and individuals interested in vector calculus, particularly those learning about coordinate transformations and vector decomposition in physics and mathematics.

perplexabot
Gold Member
Messages
328
Reaction score
5
Hello all.
I am trying to change:
E = (1/r) ar

To rectangular coordinate system. Where ar is a unit vector.

So I know r = √(x^2 + y^2)
i also think ar = ax+ay, where ax and ay are unit vectors along the x-axis and y-axis respectively.

So that would give me: E = (1/√(x^2 + y^2)) (ax + ay) = ax/√(x^2 + y^2) + ay/√(x^2 + y^2)... Which is wrong.

The correct answer is: (x ax)/(x^2 + y^2) + (y ay)/(x^2 + y^2)

So what am I doing wrong? Thank you.
 
Physics news on Phys.org
perplexabot said:
i also think ar = ax+ay, where ax and ay are unit vectors along the x-axis and y-axis respectively.

Yeah, adding two unit vectors doesn't usually give you another unit vector. This is the big issue.
 
Oh, that makes a lot of sense. Thank you. So I would have to divide by the magnitude of their sums. I will try and figure that out.
 
perplexabot said:
Oh, that makes a lot of sense. Thank you. So I would have to divide by the magnitude of their sums. I will try and figure that out.

Ok, I tried figuring this out but with negative outcome. Here is what I did:

ar = (ax + ay) / |ax + ay| = (ax + ay) / √(x^2 + y^2)

So: E = 1/r ar = (1 / √(x^2 + y^2)) * [(ax + ay) / √(x^2 + y^2)] = (ax + ay) / (x^2 + y^2)

Still, this is wrong. What did I muck up this time?
 
[(ax + ay) / √(x^2 + y^2)]
This is again an incorrect decomposition of the unit vector ar.
 
arildno said:
[(ax + ay) / √(x^2 + y^2)]
This is again an incorrect decomposition of the unit vector ar.

Ok thanks. I will give it one last try tomorrow (maybe even tonight). But if I can't do it please direct me towards the right path.

Is this part correct: ar = (ax + ay) / |ax + ay|
?
 
No.
You need to learn how to decompose vectors correctly.

We have:
ar=(x*(ax)+y*(ay))/r
 
Thank you.
 
So you multiply x with ax so that ax points in the direction of x, am I right? Same with y and ay?
 

Similar threads

Undergrad Problem with calculating projections of curl using rotation of contour
  • · Replies 9 ·
Replies
9
Views
2K
High School Why Does ln(y) = ln(bεax) Cause Issues in Derivation?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Transform from polar to cartesian
  • · Replies 2 ·
Replies
2
Views
2K
Graduate How to take the spatial derivative of quaternions
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Curl operator for time-varying vector
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Simple dot product in polar coordinates
  • · Replies 20 ·
Replies
20
Views
7K
High School Tangents to a curve as functions of different variables
  • · Replies 5 ·
Replies
5
Views
2K
MHB -10.1.1 write polar to rectangular
  • · Replies 3 ·
Replies
3
Views
2K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
How do we write a sinusoidal solution to a 2nd order DE as a sum of exponentials raised to complex roots?
  • · Replies 2 ·
Replies
2
Views
3K