Convert a cylindrical coordinate vector to cartesian coordinates

In summary, the given cylindrical coordinate vector \overrightarrow{A} can be converted to a Cartesian vector using the equations A_x, A_y, and A_z. After substituting the definitions for cos\,\phi and sin\,\phi, the final Cartesian vector is expressed as \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
  • #1
VinnyCee
489
0

Homework Statement



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]



Homework 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]



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?
 
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  • #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!
 

1. What is the formula for converting a cylindrical coordinate vector to cartesian coordinates?

The formula for converting cylindrical coordinates (r, θ, z) to cartesian coordinates (x, y, z) is:
x = r cos(θ)
y = r sin(θ)
z = z

2. How do you determine the values of r and θ in a cylindrical coordinate vector?

The value of r is the distance from the origin to the point in the xy-plane, and θ is the angle between the positive x-axis and the line connecting the origin to the point. This angle is measured counterclockwise from the positive x-axis.

3. Can a cylindrical coordinate vector have negative values?

Yes, a cylindrical coordinate vector can have negative values. The r value can be negative if the point is in the negative direction from the origin, and the θ value can be negative if the angle is measured clockwise from the positive x-axis.

4. What are the limitations of converting a cylindrical coordinate vector to cartesian coordinates?

The main limitation is that the conversion only works for points in 3-dimensional space. It cannot be used for higher dimensions. Additionally, the conversion assumes that the cylindrical coordinate system is aligned with the cartesian coordinate system, with the z-axis perpendicular to the xy-plane.

5. Can you convert a cartesian coordinate vector to cylindrical coordinates?

Yes, it is possible to convert a cartesian coordinate vector (x, y, z) to cylindrical coordinates (r, θ, z) using the following formulas:
r = √(x² + y²)
θ = tan⁻¹(y/x)
z = z

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