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Homework Help: Identify these surfaces- quick vector question

  1. Oct 29, 2011 #1
    1. The problem statement, all variables and given/known data
    Identify the following surfaces:
    i) r.u=L
    ii) r.u=mlrl for -1[itex]\leq[/itex]m[itex]\leq[/itex]1
    where k, L, m are fixed scalars and u is a fixed unit vector.


    2. Relevant equations



    3. The attempt at a solution
    The first one is in the same form as the equation of a plane, but u is not necessarily the normal, so I'm confused. For the second one, I have no idea.
     
  2. jcsd
  3. Oct 29, 2011 #2
    Is part i just an ordinary plane?
     
  4. Oct 29, 2011 #3

    I like Serena

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    Hi Lucy Yeats! :smile:

    Yes.

    r.u is the distance of point r to the surface perpendicular to u through the origin.

    With r.u=L you get all points with distance L to this surface, which is again a surface.
     
  5. Oct 29, 2011 #4

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    As for part ii, you would get all points with distance m|r| to the plane normal to u and through the origin.

    Let's start with m=0. What is it?
    Now m=1: can you say in words which points you get?
     
  6. Oct 29, 2011 #5
    Since m is a fixed scalar, I don't think it changes. However, the modulus of r is increasing with distance from the origin. So do you get a kind of 3d parabola/ bowl shaped surface??
     
  7. Oct 29, 2011 #6

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    Ah, you're ahead of me (but no, it is not a bowl :wink:).

    With m=0, you'd get r.u=0 which is a plane through the origin.

    With m=1, you'd get all points r with a distance to the plane that is equal to the distance of r to the origin.
    Which points would that be?
     
  8. Oct 29, 2011 #7
    All the points halfway between the origin and the surface?
     
  9. Oct 29, 2011 #8

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    No. The surface we're talking about contains the origin.
    So there's no such thing as halfway.

    Perhaps you can make a drawing in 2 dimensions.
    Instead of a plane we'll have a line, but the principle remains the same.
    Let u=(0,1).
    What is the "plane" in this example?
    Can you find a point that has an equal distance to the origin as to the line representing the "plane"?
     
  10. Oct 29, 2011 #9

    ehild

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    How do you get r˙uusing the length of the vectors and the angle they enclose? Try to use this formula.

    ehild
     
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