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Projection and Reflection of Vector WRT plane

  1. May 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Given a plane [itex]\Pi[/itex] with normal [itex]n=i-2j+k[/itex] and a vector [itex]v=3i+4j-2k[/itex] calculate the projection of [itex]v[/itex] onto [itex]\Pi[/itex] and the reflection of [itex]v[/itex] with respect to [itex]\Pi[/itex].

    3. The attempt at a solution

    I need to check that I'm doing this is right.

    I think I need [itex]v - (v \cdot n)n = 3i+4j-2k - 7(i-2j+k) = -4i +18j-9k[/itex]

    And for the refection:

    [itex]v - 2(v \cdot n)n = 3i+4j-2k - 14(i-2j+k) = -11i +32j-16k[/itex]

    Are those correct?
     
  2. jcsd
  3. May 14, 2012 #2

    vela

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    You made a sign error.

    The projection of v onto the plane should be perpendicular to the plane, right? So what should your answer dotted with ##\vec{n}## equal? That's how you can check your answer.


    Same sign error.
     
  4. May 14, 2012 #3
    Makes sense, thanks for the quick response.

    I get [itex]10i-10k+5k[/itex] which, when dotted with [itex]n[/itex] obviously gives 0.

    Correcting the sign error for the second yields [itex]17i-24j+12k[/itex].

    Thanks for your help
     
  5. May 14, 2012 #4

    vela

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    That's not correct either. You need to normalize the normal vector before you use it in your formulas.
     
  6. May 14, 2012 #5
    Ah, I see.

    So instead I use [itex]\hat{n}=\frac{1}{\sqrt{6}}(1i-2j+1k)[/itex]

    and that gives [itex]Pv=\frac{1}{6}(25i+10j-5k)[/itex].

    And [itex]Tv=\frac{1}{6}(32i-4j+2k)[/itex].

    Is that right now?
     
  7. May 14, 2012 #6

    vela

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    Yes, those are correct.
     
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