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Vector transformation, cylindrical to Cartesian

by tomwilliam
Tags: cartesian, cylindrical, transformation, vector
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tomwilliam
#1
Mar11-12, 11:36 AM
P: 128
1. The problem statement, all variables and given/known data
I have a result which is in the form (cylindrical coordinates):

$$ A\boldsymbol{e_{\theta }}=kr\boldsymbol{e_{\theta }} $$

And I have to provide the answer in cartesian coordinates.


2. Relevant equations

I know that the unit vectors:

$$ \boldsymbol{\hat{\theta} }=\begin{bmatrix}-sin\ \theta
\\
cos\ \theta
\end{bmatrix} $$

and that
$$ r=\sqrt{x^{2}+y^{2}} $$


3. The attempt at a solution

$$ kr\boldsymbol{e_{\theta }} =k\left (\sqrt{x^{2}+y^{2}}\right ) $$

$$ \left (-sin\left(tan^{-1}\left(\frac{y}{x}\right ) $$
$$ \right )\boldsymbol{e_{x}} $$
$$ +cos\left (tan^{-1}\left (\frac{y}{x}\right )\boldsymbol{e_{y}} \right )\\ $$

I can't seem to get further than this. I don't know if I've made a mistake, or whether there is some trig identity that can help me simplify further, but I know the final answer and it is much simpler.
Any help much appreciated.
Thanks in advance
P.S. Why does the latex break down when the equation is too long?
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