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Vector Calc: Ortho Projections

  1. May 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the projection of P in the direction of Q and the component of P orthogonal to Q.

    P=i-3j+k Q=-i+2j+5k



    2. Relevant equations
    ProjQ(P)={(P*Q)/(Q*Q)}Q

    OrthQ(P)=P-ProjQ(P)


    3. The attempt at a solution

    First I get P=(1,-3,1) and Q=(-1,2,5)

    ProjQ(P)
    =(-2/30)*(-1,2,5)
    ={(1/15),(-2/15),(-1/3)}

    That's correct, as far as the book is concerned.

    Now, finding the orthogonal part is where I get held up.

    I do this:
    OrthQ(P)=
    =(1,-3,1)-{(1/15),(-2/15),(-1/3)}
    ={14/15,(47/15),(4/3)}

    Which is wrong. The book says the correct answer is: {(74/15),(32/15),(-2/3)}.

    I'm at a complete loss as to how they got this. Any help is much appreciated.
     
    Last edited: May 13, 2010
  2. jcsd
  3. May 13, 2010 #2

    gabbagabbahey

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    Your formula for [itex]\text{Proj}_\textbf{Q}(\textbf{P})[/itex] should have [itex]\textbf{Q}\cdot\textbf{Q}[/itex] in the denominator, not [itex]\textbf{P}\cdot\textbf{P}=11[/itex], although you seem to have done the calculations correctly.

    I think the answer given in the back of your text for the second part is wrong, and they switched to using [itex]\textbf{P}=5\textbf{i}+2\textbf{j}-\textbf{k}[/itex] by mistake in that 2nd part to get their (incorrect) answer.
     
    Last edited: May 13, 2010
  4. May 13, 2010 #3

    jbunniii

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    I assume you mean

    ProjQ(P)={(P*Q)/(Q*Q)}Q

    which must be what you used since you got the right answer for ProjQ(P).

    For OrthQ(P), I get (14, -43, 20)/15, which doesn't match either your answer or the book's.
     
  5. May 13, 2010 #4
    Ah, yes. It should be QQ in the denominator rather than PP. I just translated it incorrectly to the forum, I've been working on this for a while now and am getting frustrated.

    But if they're using P=5i+2j-k...that's not in the problem. It appears the textbook is incorrect?
     
  6. May 13, 2010 #5

    gabbagabbahey

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    Yes, the textbook is incorrect. However,as jbunnii pointed out, your answer is also a little off (pay close attention to your negative signs)
     
  7. May 13, 2010 #6
    Alright, I realize now what I did was incorrect. OrthoP should be = (14, -43, 20)/15, I forgot that there was a -1/3 in ProjP which is why I didn't get 20/15 and added the -2 wrong to 45.

    However, that's still not what's in the back of the book. Argh. I guess I can take my case to the prof in the morning. Thanks guys.
     
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