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Converting from Cylindrical to Cartesian

  1. Jan 24, 2009 #1
    1. This is not a question from the book but i think if i can get the answer to this it will clear the idea i am confused about

    can i covert a cylindrical vector such as

    P(1 ar, 1a[itex]\theta[/itex])

    into cartesian

    after using the matrix

    i got

    Px = cos [itex]\theta[/itex] - sin [itex]\theta[/itex]
    Py = sin [itex]\theta[/itex] + cos [itex]\theta[/itex]

    Now these values of phi are dependent on a point right? So i cannot find numbers until the question asks for the conversion at a specific point...

    for example
    at point (1,2)

    then cos [itex]\theta[/itex] = x/sqrt(x^2+y^2)


    OR DOES [itex]\theta[/itex] = 1 from

    the vector P(1 ap, 1 a[itex]\theta[/itex])
    Last edited: Jan 24, 2009
  2. jcsd
  3. Jan 24, 2009 #2


    Staff: Mentor

    What is this? I don't understand ap and aphi. In a cylindrical coordinate system you have three coordinates: r, [itex]\theta[/itex], and z. To convert into Cartesian coordinates, you have:
    x = r * cos([itex]\theta[/itex])
    y = r * sin([itex]\theta[/itex])
    z = z
  4. Jan 24, 2009 #3
    yes that is to covert specific coordinates but to covert vectors you need to use the following for cylindrical to cartesian

    Px = cos [itex]\theta[/itex] - sin [itex]\theta[/itex]
    Py = sin [itex]\theta[/itex] + cos [itex]\theta[/itex]

    and obviously

    Pz = z

    since that does not change

    but what i was asking is in 2d to make it even simpler because I am very confused

    So I have a vector in cylindrical coordinate system

    P ( 1 ar, 1 a[itex]\theta[/itex])
  5. Jan 24, 2009 #4


    Staff: Mentor

    I'm still not understanding your notation: P ( 1 ar, 1 a[itex]\theta[/itex]).
    What do ar and a[itex]\theta[/itex] mean?
  6. Jan 24, 2009 #5
    they are unit vectors in the R direction and in the Theta direction

    they are just representing vectors in Cylindrical System
    for example

    In cartesian system a vector can be represented like

    1 ax + 2 ay or 1 i + 2 j etc. where ax = i and ay = j are unit vectors

    similarly my notation i guess is confusing but its just the unit vectors in the respective coordinate system

    Cylindrical (r theta and z)
  7. Jan 24, 2009 #6


    Staff: Mentor

    Maybe I'm missing something, but this doesn't make any sense to me. Considering 2D cartesian and polar coordinates for the moment, the point A(1, 1) in Cartesian coordinates can also be used to define a vector, OA, which can be represented as 1i + 1j. The same point in polar coordinates is ([itex]\sqrt{2}[/itex], [itex]\pi/4[/itex]). I don't see how you can break up this vector into radius and angle components. I understand the idea of a unit vector in the direction of r, but I completely don't understand the concept of the direction of an angle, and particularly a unit vector with that direction. How do you define the direction of an angle? The only possibilities I can think of would be the direction of the starting ray (the x axis) or the ending ray (which is the direction along which r lies).

    Are you talking about radial and tangential components of the vector?
    Why complicate things in what seems like a needless way with "ax" and "ay"? What is the purpose of "a" in these expressions?
  8. Jan 24, 2009 #7
    I dont know I'm just learning this that is what I am confused about :(..

    here is what it says in my textbook


    maybe you can make some sense out of the VECTOR in the cylindrical coordinate system and explain briefly to me :(
  9. Jan 25, 2009 #8


    Staff: Mentor

    OK, now that makes sense. Glad you included the figure to help me out.
    a[itex]\phi[/itex] as shown in the figure is a unit vector that is tangential, while a[itex]\rho[/itex] is a unit vector in the radial direction.

    The notation that is used is nonstandard from what I remember. The cylindrical coordinates for a point are (r, [itex]\theta[/itex], z) where [itex]\theta[/itex] is the same angle as that shown as [itex]\phi[/itex] in the figure.

    The spherical coordinates for a point are ([itex]\rho[/itex], [itex]\theta[/itex], [itex]\phi[/itex]), where [itex]\rho[/itex] measures the distance from the origin to the point, [itex]\theta[/itex] is the same as used in polar or cylindrical coordinates, and [itex]\phi[/itex] is the angle measured from the positive z-axis down to the line segment from the origin to the point. The figure mixes up [itex]\theta[/itex] and [itex]\phi[/itex]

    To answer your original question,
    Yes, but let's call it theta for the sake of cylindrical coordinates. The value of theta and therefore a[itex]\theta[/itex] depend on the point, but theta and a[itex]\theta[/itex] are different things.
    I don't believe so. |a[itex]\theta[/itex]| = 1, but [itex]\theta[/itex] doesn't have to be 1.
  10. Jan 25, 2009 #9
    Ok thank you :),
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