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Vector Problem

  1. Oct 29, 2012 #1
    Vector Problem plz help

    1. The problem statement, all variables and given/known data
    To prove using the vector method for any triangle ABC the statement c=acosB+bcosA holds.


    2. Relevant equations[/

    A.B=abcosα
    3. The attempt at a solution

    Frankly, I don't know how to go about it. At first it seemed easy but I just coudn't somehow use the dot product twice to get acosB and bcosA. It would be greatly appreciated if you guy gave me some ideas that would help me this proof..
     
  2. jcsd
  3. Oct 29, 2012 #2

    mfb

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    Re: Vector Problem plz help

    If A and B are vectors, what are cos(A) and cos(B) supposed to be?
    a,b,c are the lengths of the sides A, B, C?

    That equation looks strange.
     
  4. Oct 29, 2012 #3
    Re: Vector Problem plz help

    No, a,b and c are supposed to be the position vectors of A, B and C. I am first supposed to get the sum of two vectors like for example AB+BC=AC and expand it from here on.
     
  5. Oct 29, 2012 #4

    mfb

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    Re: Vector Problem plz help

    Ok, so AB, BC, AC are vectors corresponding to the sides of your triangle.
    But what are cos(A) and cos(B)?
     
  6. Oct 29, 2012 #5
    Re: Vector Problem plz help

    Here, acosB= BCcosB and bcosA=ACcosA. acosB and bcosA are also supposed to be two vectors of triangle ABC.
     
  7. Oct 29, 2012 #6

    mfb

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    Re: Vector Problem plz help

    I don't think that notation makes any sense.
     
  8. Oct 29, 2012 #7
    Re: Vector Problem plz help

    Okay, Could you give me the starting point of this problem?? CosB is apparently a/c and cosA=b/c.
     
  9. Oct 29, 2012 #8

    mfb

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    Re: Vector Problem plz help

    The problem is that I don't even understand your problem.

    Can you show the original source? Link, photo, screenshot, whatever?
    If that is not possible, please copy the exact problem statement, including all definitions and anything else which belongs to the problem.

    c is a vector, you cannot divide by vectors.
    And even if I replace c by |c|: that would imply c^2=a^2+b^2 for any triangle, which is wrong.
     
  10. Oct 29, 2012 #9
    Re: Vector Problem plz help

    It's actually from my textbook and no descriptive information is given regarding this problem which is why i am at a total loss. The original problem is Prove by vector method in any triangle ABC 1) cosC= a^2+b^2-c^2/2ab
    2)c=acosB+bcosA and the only example regarding these problems in the book given is cosA=b^2+c^2-a^2/2bc
    BA+AC=BC
    or, (BA+AC)(BA+AC)=BC.BC
    or, BA^2+AC^2+2BA.AC=BC^2
    or, c^2+b^2+2|AC|BA|cos(180-A)
    or, c^2+b^2-2bccosA+a^2
    or, cosA= b^2+c^2-a^2/2bc
    Hope, you wouldn't mind me writing this long post as you yourself told me to present all the definitions and info pertaining to this prob.
     
    Last edited: Oct 29, 2012
  11. Oct 29, 2012 #10

    mfb

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    Re: Vector Problem plz help

    That post is not long ;).
    Apparently A, B, C are the angles in the triangle. Strange, but ok.

    Just by renaming a,b,c and A,B,C, you get cos(C)= a^2+b^2-c^2/(2ab)

    Concerning question 2, did you draw a sketch? Try to mark a*cos(B) on the c-side, beginning at the point where the angle B is. Is the other point on c special in some way?
     
  12. Oct 29, 2012 #11
    Re: Vector Problem plz help

    Sorry but I didn't understand what you meant by your question . small a, b, and c are the position vectors and capital A, B and C are the angles of the triangle. Yes the notations are a bit confusing. Could you just tell me if the dot product will help me in any way in this proof.
     
    Last edited: Oct 29, 2012
  13. Oct 29, 2012 #12

    mfb

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    Re: Vector Problem plz help

    The dot product can help. But I really suggest to look at the geometric interpretation with a sketch first.
     
  14. Oct 30, 2012 #13
    Re: Vector Problem plz help

    I have sent the sketch in the attachment and see if it's okay.
     

    Attached Files:

  15. Oct 30, 2012 #14

    mfb

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    Re: Vector Problem plz help

    Oh, an additional catch: The equation requires that a,b,c there are the magnitudes of the vectors. With vectors, it is wrong.
    ##|\vec{c}|=|\vec{a}|\cos(B)+|\vec{b}|\cos(A)##

    It is easier to see if you draw a and b such that c=a+b. Just try to modify that equation to get dot products inside.
     
  16. Oct 30, 2012 #15
    Re: Vector Problem plz help

    I think I finally have some idea about solving it. See, if my process is okay or not. I actually have some difficulty using the vector symbols in computers. But, I am first starting off with vectors.

    c=a+b
    or, BA=BC+CA
    or, BA.BA= BC.BA+CA.BA
    or, |BA|^2= |BC||BA|cos(180-B) + |CA||BA|cos(180-A)
    or, |c|^2= |c||a|cosB +|c||b|cosA
    or, |c|= |a|cosB + |b|cosA
     
  17. Oct 30, 2012 #16

    mfb

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    Re: Vector Problem plz help

    Without the "180-", but that is good.
     
  18. Oct 30, 2012 #17
    Re: Vector Problem plz help

    Do you think including the "180-" is quite superficial and is it better to exclude it from my solution.
     
  19. Oct 30, 2012 #18

    mfb

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    Re: Vector Problem plz help

    I don't see where it should come from, and removing it like you did would switch the sign.
     
  20. Oct 31, 2012 #19
    Re: Vector Problem plz help

    So, thanks for all the help that you have provided in solving this problem and I finally have gained a better understanding that will hopefully help me in solving these problems.
     
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