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Equation in R^4, R^5, and R^6

  1. Sep 8, 2008 #1
    1. The problem statement, all variables and given/known data
    What does the equation [tex]x^2 -2x + y^2 -2y + z^2 -2z + t^2 -2t + 4 = 0[/tex]
    represent in [tex]\Re^4[/tex]? What does it represent in [tex]\Re^5[/tex]? What does it represent in [tex]\Re^6[/tex]?

    2. Relevant equations
    Complete the square.

    3. The attempt at a solution
    After completing the square (letting 4 = 1 + 1 + 1 + 1), the equation becomes

    [tex](x-1)^2 + (y-1)^2 + (z-1)^2 + (t-1)^2 = 0[/tex]

    In [tex]\Re^4[/tex], the equation represents the coordinate (1,1,1,1). I have a hard time visualizing the cases for [tex]\Re^5[/tex] and [tex]\Re^6[/tex]. I figure that it should be easier to visualize knowing that in [tex]\Re^4[/tex], we have a fixed point.

    Of course I have deduced that we will get a line and a plane in [tex]\Re^5[/tex] and [tex]\Re^6[/tex], respectively, by looking at smaller cases. But what is the exact configuration and how do I see it?
  2. jcsd
  3. Sep 9, 2008 #2


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    Homework Helper

    … don't worry …

    Hi snipez90! :smile:

    "a line and a plane" is the answer! :smile:

    If you want to give the exact equation, the line, for example, would be x=y=z=t=1 (and w = anything). :wink:
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