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Geometry problem involving dot/cross product

  1. Sep 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Let r be a line and pi be a plane with equations
    r: P + tv
    pi: Q + hu + kw (v, u, w are vectors)
    Assume v · (u x w) = 0. Show that either r ∩ pi = zero vector or r belongs to pi.

    2. Relevant equations
    n/a


    3. The attempt at a solution
    I get the basic idea behind it but I'm not sure if my "solution" is formally good enough. I know that cross product gives you a normal to a plane, which is u x w here. I also know that the dot product = 0 means that they are perpendicular to each other.. so v and (u x w) are perpendicular to each other. If you draw it out, you can clearly see that r must be either in pi or out of pi. Is this good enough to warrant a good mark? Thanks!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 28, 2009 #2

    lanedance

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    Homework Helper

    if r is parallel to pi, but not "in" pi their intersection will not be the zero vector, it will be the empty set

    try the separate cases when P is "in" pi, and when P is not "in" pi and try and see if you can show whether any arbitrary point on r is in, or not in P, knowing that v = a.(uXw) where a is some non-zero constant
     
  4. Sep 29, 2009 #3
    you are right, my mistake


    I can show that with intuition and a sketch of plane/line but not sure how I should go about proving it formally.. maybe this Q is that simple and I'm overreacting
     
  5. Sep 29, 2009 #4

    lanedance

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    as you know
    v.(u x w) = 0

    then
    v = au + bw
    for constants a & b - why?

    equation of you line is P + vt

    now, i haven't tried, but using your equation of a line & the equation of a plane, considering the following cases, should show what you want:

    Case 1 - P is a point in pi. Now show any other point on the line, using the line equation, satisfies the equation defining pi.

    Case 2 - P is not a point in pi. Now show any other point on the line, using the line equation, does not satisfy the equation defining pi.
     
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