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Homework Help: Geometry question in 3 space.

  1. Aug 26, 2008 #1
    1. The problem statement, all variables and given/known data

    (a) Find the vector equation of the plane through the points (2,−1, 0) and (−5,−3, 1)
    that is parallel to the line joining the points (3, 5,−1) and (0, 3,−2).
    (b) Find the parametric equations of the straight line though the origin that is perpendicular
    to this plane, and find where it intersects the plane.

    2. Relevant equations

    [tex]\hat{n} \cdot (r~-~r_{0}) = 0[/tex]

    3. The attempt at a solution
    Ok, I found two vectors, on joining the points (2,−1, 0) and (−5,−3, 1) and another joining the points (3, 5,−1) and (0, 3,−2), seeing as they are parallel they will share the same normal vector [tex]\hat{n}[/tex].

    Then using [tex]\hat{n} \cdot (r~-~r_{0}) = 0[/tex] with r being (-5,-3, 1) and [tex]r_{0}[/tex] being (2,-1,0). To get the vector form of the equation plane.

    but I am not exactly sure what points to use at this stage.

    and I haven't attempted b as yet.

  2. jcsd
  3. Aug 26, 2008 #2


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    Homework Helper

    (a) Use a vector operation to get the normal vector.

    (b)For this you need the normal vector from a). Note that [tex]k\mathbf{n}[/tex] for some real value of k will give you a point on the plane. Think about what the line corresponding to this position vector means.
  4. Aug 26, 2008 #3
    ok, are my methods correct, so far?

  5. Aug 27, 2008 #4
  6. Aug 27, 2008 #5


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    Those two vectors are certainly NOT parallel. Why would you think they are? The vector connecting the first two points is <7, 2, -1> and, connecting the last two, <3, 2, 1>. Since one is not a multiple of the other, they are not parallel.

  7. Aug 27, 2008 #6
    Ok yep, I assumed that because <3, 2, 1> was parallel to the plane it would be parallel to a vector lying on that plane <7, 2, -1>.

    So where exactly do I go from here?
  8. Aug 27, 2008 #7


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    It isn't parallel, which is a good thing since otherwise that vector operation wouldn't work! For any plane there is an infinite number of vectors which are parallel to it; just think of rotating an arrow on a board by 360 degrees, every possible vector corresponding to the arrow direction is parallel to the plane. It is precisely that it is non-parallel that you can apply the said vector operation to get the normal vector.
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