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Converting cartesian to parametric equation

  1. Jun 24, 2009 #1
    converting cartesian to parametric equation R3


    I can't convert cartesian to parametric equation this equation 3x-y+4z-6=0
    In example is given only 3x-y+4z-6=0 and says to convert it to parametric form ?

    how this can be done ?
    Last edited: Jun 24, 2009
  2. jcsd
  3. Jun 24, 2009 #2


    Staff: Mentor

    So that I don't completely answer your question for you, let me show you how it works for a different problem. Suppose your equation was x + 2y - z + 2 = 0. This equation represents a plane in R3, as does your equation.

    Solving for x gives
    x = -2y + z - 2
    We're going to need a set of equations for x, y, and z, so here are two more:
    y = y
    z = z
    The last two equations are obviously and trivially true.

    Here's what we have:
    Code (Text):

    x = -2y + z - 2
    y = y
    z =      z

    From this, we can see that any point (x, y, z) in the plane can be written in terms of two parameters y and z. Namely, (x, y, z) = y*(-2, 1, 0) + z*(1, 0, 1) + (-2, 0, 0).

    If you like, you can use different letters for the parameters, say r and s, so that you have
    (x, y, z) = r*(-2, 1, 0) + s*(1, 0, 1) + (-2, 0, 0).

    If you look at this in terms of vectors, the vector (-2, 0, 0) takes you from the origin to the point (-2, 0, 0) in the plane, and the other two vectors take you from that point to any other in the plane.

    You can apply the same thinking to your problem. Hope that helps.
  4. Jun 24, 2009 #3


    User Avatar
    Science Advisor

    Note that this is a plane. That is, it is two dimensional and so will require two parameters (unlike a line or curve that is one dimensional and so requires only one parameter). What Mark44 did was use y and z as parameters. If you don't like that, that is, if you prefer to use, say, s and t, as parameters, just replace y and z by that.

    That is, if x= -2y+ z- 2 then x= -2s+ t- 2, y= s, z= t are parametric equations for the plane.

    In general, if you can solve an equation, in x, y, and z, representing a surface, for any one of the variables, you can use the other two as parameters.
  5. Jun 24, 2009 #4
    thank you for your answers,
    now I understand how to express it with parameters
    but this part, (x, y, z) = r*(-2, 1, 0) + s*(1, 0, 1) + (-2, 0, 0)
    i don't understand
    how (-2,1,0) and (1,0,1) vectors are acquired !

    thanks again
  6. Jun 24, 2009 #5


    Staff: Mentor

    From here:
    Code (Text):

    x = -2y + 1z - 2
    y = 1y + 0z
    z = 0y + 1z
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