Polar to Cartesian Unit Vectors in 2D

Click For Summary

Homework Help Overview

The discussion revolves around expressing the Cartesian unit vectors x-hat and y-hat in terms of the polar unit vectors r-hat and phi-hat. Participants are exploring the relationships between these vector representations in a two-dimensional context.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to derive expressions for x-hat and y-hat using the provided equations for r-hat and phi-hat. There are questions about the complexity of the resulting expressions and the methods for determining coefficients α and β in the proposed relationship.

Discussion Status

The discussion is active, with participants providing guidance on using dot products to find the coefficients needed for the conversion. There are also inquiries about the origins of the equations and suggestions to analyze specific examples to deepen understanding.

Contextual Notes

Some participants express confusion regarding the source of the equations and the rationale behind their selection, indicating a need for clarification on assumptions or definitions related to the problem.

leahc
Messages
2
Reaction score
0

Homework Statement


Solve for the unit vectors x-hat and y-hat in terms of r-hat and phi-hat.


Homework Equations


r-hat=cos(phi)x-hat+sin(phi)y-hat
phi-hat=cos(phi)y-hat-sin(phi)x-hat,


The Attempt at a Solution


I have been working on this for a really long time, and I keep getting a really complicated expression for x-hat, with everything over cos(phi)^2. That seems wrong, and I can't figure out how to solve it from only those two given equations.
 
Physics news on Phys.org
Let ##\hat{x} = \alpha\hat{r} + \beta\hat{\phi}##.
Compute ##\hat{x}.\hat{r}## etc.
 
I understand that, but how do I compute α and β?
 
Haruspex already told you how. That's a dot product in case you didn't recognize it.
 
Where did you get those equations? Why did you choose them? Take one example of an x-y unit vector and calculate r and phi - plotting it might help. Then take a different example and do likewise. Notice anything?
 

Similar threads

  • · Replies 19 ·
Replies
19
Views
4K
  • · Replies 25 ·
Replies
25
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 6 ·
Replies
6
Views
3K
Replies
5
Views
3K
Replies
11
Views
2K
  • · Replies 3 ·
Replies
3
Views
4K
Replies
9
Views
3K