Resolve the cartesian unit vectors into their cylindrical components

In summary, the conversation is about resolving cartesian unit vectors into their cylindrical components using scale factors. The person is having trouble with this and is looking for clarification. The answer involves three vectors, \vec{e_{r}}, \vec{e_{\theta}}, and \vec{e_{z}}, which can be found by finding the normals to surfaces created when r, \theta, and z are constant.
  • #1
JasonPhysicist
13
0

Homework Statement


The problem is :''Resolve the cartesian unit vectors into their cylindrical components(using scale factors)




The Attempt at a Solution


It's simple to do the inverse(resolving cylindricl unit vectors into cartesian components),but I'm having some ''trouble'' with the above problem.
I know the answer is:

[tex]x=\rho\cos\varphi - \varphi\sin\varphi[/tex]
[tex]y=\varphi\sin\varphi + \varphi\cos\varphi[/tex]
[tex]z=z[/tex]


Could someone shed some light?Thank you.





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  • #2
I think you're forgeting the vectors in the answer that you know. If you're converting the i,j,k (or x,y,z) unit vectors into their equivalent vectors in cylinderical coordinates, your answer will be in terms of 3 vectors, [tex]\vec{e_{r}},\vec{e_{\theta}},\vec{e_{z}}[/tex] (they might be presented differently in different texts though.)

The way to get to these base vectors is to look at the normals to the surfaces that are created when theta, r, and z are fixed constant. For example if you wish to find [tex]\vec{e_{r}}[/tex]
you need to find normals to the surfaces what are created when r is constant -- that is cylindrical shells. You can do this by observation, or by simply noting the fact that a cylinder is:
[tex] x^{2}+y^{2} = r^{2} [/tex]
and then simply finding a normal to it:
[tex]\vec{e_{r}} = cos(\theta)\vec{i}+sin(\theta)\vec{j}[/tex]
you can do this for all of the vectors, and then solve the system of 3 equations to find the 3 vectors you are looking for
 

1. What are cartesian unit vectors?

Cartesian unit vectors are the three basic unit vectors used in the Cartesian coordinate system to define a point in three-dimensional space. They are called the x, y, and z unit vectors, and are denoted by i, j, and k, respectively.

2. What is the relationship between cartesian and cylindrical coordinates?

Cylindrical coordinates are an alternate way of representing points in three-dimensional space, using a combination of radial distance, angle, and height. The x and y components in cartesian coordinates correspond to the radial distance and angle in cylindrical coordinates, while the z component remains the same.

3. How do you resolve cartesian unit vectors into their cylindrical components?

To resolve cartesian unit vectors into their cylindrical components, you can use the following equations:

  • r = √(x^2 + y^2) (radial distance)
  • θ = tan^-1(y/x) (angle)
  • z = z (height)

These equations will give you the values for r, θ, and z in cylindrical coordinates.

4. What are some applications of resolving cartesian unit vectors into cylindrical components?

Resolving cartesian unit vectors into cylindrical components is useful in fields such as physics, engineering, and mathematics. It allows for a more efficient and accurate representation of points in three-dimensional space, and is particularly useful in problems involving circular or cylindrical objects.

5. Are there other coordinate systems besides cartesian and cylindrical?

Yes, there are many other coordinate systems used in different fields of science and engineering. Some examples include spherical coordinates, polar coordinates, and curvilinear coordinates. Each coordinate system has its own unique set of unit vectors and equations for representing points in three-dimensional space.

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