Convert a cylindrical coordinate vector to cartesian coordinates

[VinnyCee]

Homework Statement

Convert the following cylindrical coordinate vector to a Cartesian vector:

\overrightarrow{A}\,=\,\rho\,z\,sin\,\phi\,\hat{\rho}\,+\,3\,\rho\,cos\,\phi\,\hat{\phi}\,+\,\rho\,cos\,\phi\,sin\,\phi\,\hat{z}

Homework Equations

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

Following the steps in the above equation...

A_y\,=\,A_{\rho}\,sin\,\phi\,+\,A_{\phi}\,cos\,\phi

A_z\,=\,A_z

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

cos\,\phi\,=\,\frac{x}{\rho}

sin\,\phi\,=\,\frac{y}{\rho}

\rho^2\,=\,x^2\,+\,y^2

The Attempt at a Solution

Using the above equations for A_x, A_y and A_z, I get:

A_x\,=\,\rho\,z\,cos\,\phi\,sin\,\phi\,-\,3\,\rho\,cos\,\phi\,sin\,\phi

A_y\,=\,\rho\,z\,sin^2\,\phi\,+\,3\,\rho\,cos^2\,\phi

A_z\,=\,\rho\,cos\,\phi\,sin\,\phi

Now combine into a vector...

\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)

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

\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)

\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)

\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)

Does that seem right, or is there more simplification that can be done?

 

[VinnyCee]

It looks right, but I just want to make sure it is before going on to the rest of the problems like this one!