# Convert a cylindrical coordinate vector to cartesian coordinates

## 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?