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Thread: Linear Algebra:Planes help

  1. #1
    Jun 2011

    Unhappy Linear Algebra:Planes help

    The example is :
    Convert the scalar equation 4x − 2y + z = 2 into a parametric form.
    This is not a big  deal, usually. We have the coefficients, 4, -2 and 1, so the normal is
    (4,−2,1)=2 it’s not = 0, so there will be 2 vectors orthogonal
    to it. Any two vectors that are not in the same direction will do. The best way to make sure they aren’t in the same direction is to make them cover different variables, i.e., use only x and y for one, y and z for the other.

    For instance,

    (1,2,0) is orthogonal to (4,−2,1) as is(0,1,2)
    Those are clearly not in
    the same direction. They’ll do. Recall that we need to match the right hand side. Any
    combination of those two will not change the total on the left. We just need a single point that matches. So,
    x =1/2-->(1/2,0,0)
    z=2--> ( 0,0,2)

    So, any will do, here’s one final set


    From all this my questions are when they mention that the normal is (4,-2,1) they say that the orthogonal is (1,2,0) and (0,1,2).

    I was wondering how they got the orthogonal and how they got those two from the normal (4,-2,1)?

    My second question is how did they get the final set?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Dec 2009

    Re: Linear Algebra:Planes help

    If you want to find a parametric equation to Ax+By+Cz+D=0 follow these steps:

    1. find 3 points on the plane Ax+By+Cz+D=0 (you can take x=y=0, then find z\ Say that (A,B,C) is your normal then (A,B,C)(a,b,c)=0 )
    2. Suppose that 3 points are A, B and C. Create two vectors AB and AC.
    3. Construct your parametric equation: (point on a plane)+vec(AB)*t +vec(AC)*s
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  3. #3
    MHF Contributor

    Apr 2005

    Re: Linear Algebra:Planes help

    A much simpler way to arrive at parametric equations for a plane is to solve for one variable in terms of the other two:
    2x- y+ z= 2 is the same as z= 2- 2x+ y. So x= x, y= y, z= 2- 2x+ y are parametric equations for the plane. If you don't like the idea of using x and y as paramters, just "rename" them: x= s, y= t, z= 2- 2s+ t.
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  4. #4
    Jun 2011

    Re: Linear Algebra:Planes help

    I still don't understand what to do, so my original equation is 4x-2y+z=2 and i know the normal is 4,-2, 1 so what would i do to get it into parametric form? Are you able to give me an example of how to do this question?
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