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Convert a cylindrical coordinate vector to cartesian coordinates

  1. Sep 5, 2008 #1
    1. The problem statement, all variables and given/known data

    Convert the following cylindrical coordinate vector to a Cartesian vector:

    [tex]\overrightarrow{A}\,=\,\rho\,z\,sin\,\phi\,\hat{\rho}\,+\,3\,\rho\,cos\,\phi\,\hat{\phi}\,+\,\rho\,cos\,\phi\,sin\,\phi\,\hat{z}[/tex]



    2. Relevant equations

    [tex]A_x\,=\,\hat{x}\,\cdot\,\overrightarrow{A}\,=\,\left(\hat{x}\,\cdot\,\hat{\rho}\right)\,A_{\rho}\,+\,\left(\hat{x}\,\cdot\,\hat{\phi}\right)\,A_{\phi}\,+\,\left(\hat{x}\,\cdot\,\hat{z}\right)\,A_{z}\,=\,A_{\rho}\,cos\,\phi\,-\,A_{\phi}\,sin\,\phi[/tex]

    Following the steps in the above equation...

    [tex]A_y\,=\,A_{\rho}\,sin\,\phi\,+\,A_{\phi}\,cos\,\phi[/tex]

    [tex]A_z\,=\,A_z[/tex]

    Also, use these definitions after one completes initial conversion using the equations above...

    [tex]cos\,\phi\,=\,\frac{x}{\rho}[/tex]

    [tex]sin\,\phi\,=\,\frac{y}{\rho}[/tex]

    [tex]\rho^2\,=\,x^2\,+\,y^2[/tex]



    3. The attempt at a solution

    Using the above equations for [itex]A_x[/itex], [itex]A_y[/itex] and [itex]A_z[/itex], I get:

    [tex]A_x\,=\,\rho\,z\,cos\,\phi\,sin\,\phi\,-\,3\,\rho\,cos\,\phi\,sin\,\phi[/tex]

    [tex]A_y\,=\,\rho\,z\,sin^2\,\phi\,+\,3\,\rho\,cos^2\,\phi[/tex]

    [tex]A_z\,=\,\rho\,cos\,\phi\,sin\,\phi[/tex]

    Now combine into a vector...

    [tex]\overrightarrow{A}\,=\,\hat{x}\,\left(\rho\,z\,cos\,\phi\,sin\,\phi\,-\,3\,\rho\,cos\,\phi\,sin\,\phi\right)\,+\,\hat{y}\,\left(\rho\,z\,sin^2\,\phi\,+\,3\,\rho\,cos^2\,\phi\right)\,+\,\hat{z}\,\left(\rho\,cos\,\phi\,sin\,\phi\right)[/tex]

    Using the bottom three definitions in the Relevant Equations section above...

    [tex]\overrightarrow{A}\,=\,\hat{x}\,\left(x\,z\,\frac{y}{\rho}\right)\,+\,\hat{y}\,\left(y\,z\,\frac{y}{\rho}\,+\,3\,x\,\frac{x}{\rho}\right)\,+\,\hat{z}\,\left(x\,\frac{y}{\rho}\right)[/tex]

    [tex]\overrightarrow{A}\,=\,\hat{x}\,\left(\frac{x\,y\,z}{\sqrt{x^2\,+\,y^2}}\,-\,\frac{3\,x\,y}{\sqrt{x^2\,+\,y^2}}\right)\,+\,\hat{y}\,\left(\frac{y^2\,z}{\sqrt{x^2\,+\,y^2}}\,+\,\frac{3\,x^2}{\sqrt{x^2\,+\,y^2}}\right)\,+\,\hat{z}\,\left(\frac{x\,y}{\sqrt{x^2\,+\,y^2}}\right)[/tex]

    [tex]\overrightarrow{A}\,=\,\hat{x}\,\left(\frac{x\,y\,z\,-\,3\,x\,y}{\sqrt{x^2\,+\,y^2}}\right)\,+\,\hat{y}\,\left(\frac{y^2\,z\,+\,3\,x^2}{\sqrt{x^2\,+\,y^2}}\right)\,+\,\hat{z}\,\left(\frac{x\,y}{\sqrt{x^2\,+\,y^2}}\right)[/tex]

    Does that seem right, or is there more simplification that can be done?
     
  2. jcsd
  3. Sep 7, 2008 #2
    It looks right, but I just want to make sure it is before going on to the rest of the problems like this one!
     
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