Polar to rectangular coordinates

[perplexabot]

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.

 

[Muphrid]

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.

 

[perplexabot]

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]

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?

 

[arildno]

[(ax + ay) / √(x^2 + y^2)]
This is again an incorrect decomposition of the unit vector ar.

 

[perplexabot]

[(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|
?

 

[arildno]

No.
You need to learn how to decompose vectors correctly.

We have:
ar=(x*(ax)+y*(ay))/r

 

[perplexabot]

Thank you.

 

[perplexabot]

So you multiply x with ax so that ax points in the direction of x, am I right? Same with y and ay?