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Linear algebra: Vectors

  1. Nov 27, 2013 #1
    The Problem:
    Let O be the origin and let A, B, C be three points so that the quadrilateral OABC makes an parallelogram.
    Name (1/4){OA} a, and the diagonal {OB} b. Let P be the point that splits the side OC in the ratio 3 :2 from O.
    Write the vector {PA} as a linear combination of a and b.


    Attempt to the soulution:
    I have drawn the parallelogram and made some circumlocutions of the vectors.

    Skanna 1.jpg

    How shall I tackle this problem?
     
  2. jcsd
  3. Nov 27, 2013 #2

    PeroK

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    Can you say anything about AB and OC?
     
  4. Nov 27, 2013 #3

    haruspex

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    You've written AB = (3/2)(PO). The sign is wrong (did you mean OP?) and the ratio is wrong.
    The 2/3 in the next line is also wrong. P splits OC in the ratio 3:2, not 2:1.
     
  5. Nov 27, 2013 #4
    Yes, AB = (3/2)(PO) is wrong. It should be AB=-0C and therefore (2/3)AB=-(2/3)0C

    (2/3)0C=-P0

    Is it correct?

    It is 3:2 ratio but it maybe looks like 2:1. I have split it in three parts and 2 of them are P0. I hope that's right.
     
  6. Nov 27, 2013 #5

    haruspex

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    You're missing my point about the ratio. If I have a line of length 15 and cut it into two pieces in the ratio 3:2, what are the two lengths?
     
  7. Nov 27, 2013 #6
    So embarrassing, of course it should 5 pieces.
    3:2 of 15 is 9 and 6.

    Thanks for notice this stupid mistake!
     
  8. Nov 28, 2013 #7
    Here's my 2nd attempt.

    I'm not sure if it's right at all.


    Skanna 4.jpg
     
  9. Nov 28, 2013 #8

    haruspex

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    Looks like the right answer, but you have a sign error in the third line, which goes away in the fifth line. Maybe a transcription error?
     
  10. Nov 28, 2013 #9
    Thanks for your answer! Which sign do you mean? Is it when I change from AB=-0C to 0C=-AB

    That's the thing I'm not sure of is if AB=-0C is correct? Isn't the vector in the direction from C to 0 (C0) and therefore AB=C0 or one can write AB=-0C?
     
  11. Nov 28, 2013 #10

    haruspex

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    Your third line starts OP = -(3/5)AB. Why the minus?
    But where you use that to substitute for OP in the 4th line to get the 5th line, you seem to have used it without the minus sign.
     
  12. Nov 28, 2013 #11
    Maybe I'm blind but the minus is there all the time. In the third line: -(3/5)AB and in the fifth line -(3/5)(0B-4*0Q) and these two the same thing because I just substitute AB with (0B-4*0Q). Correct me if I'm wrong.

    I think of 0P as (3/5)AB in the opposite direction and thats why I have the minus sign. Maybe I shall see AB and 0C as the same value?
     
  13. Nov 28, 2013 #12

    haruspex

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    Yes, the mistake starts one line earlier than I noticed: AB=OC, not -OC.
    Having written (wrongly) OP = -(3/5)AB and (correctly) PA = -OP+4OQ, the logical deduction is PA = -(-(3/5)AB)+4OQ = (3/5)AB+4OQ = (3/5)(OB-4OQ)+4OQ. But you wrote -(3/5)(OB-4OQ)+4OQ, thereby correcting the earlier sign error.
     
  14. Nov 28, 2013 #13
    Okay I can see my mistake but I'm still confused.
    (3/5)(0B-4OQ)+40Q and -(3/5)(0B-4OQ)+40Q doesn't give the same answer.

    (3/5)(OB-4OQ)+4OQ= (3/5)0B+(8/5)0Q
    and
    -(3/5)(OB-4OQ)+4OQ=-(3/5)0B+(32/5)0Q
     
  15. Nov 28, 2013 #14

    haruspex

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    You made two mistakes that cancelled.
    The correct line 2, AB=OC, leads to what you posted, PA=-(3/5)(OB-4OQ)+4OQ.
    Your version, AB=-OC, should have led you to PA=(3/5)(OB-4OQ)+4OQ, but your second mistake happened to yield the correct line PA=-(3/5)(OB-4OQ)+4OQ.
     
  16. Nov 28, 2013 #15
    Now I understand. It reminds me of the musical joke:
    If you play a wrong note once it's a mistake. But if you play the wrong note twice, it's jazz.

    Two mistake and I got the answer right!


    So I hope this is correct?

    Skanna 5.jpg
     
  17. Nov 28, 2013 #16

    haruspex

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    Looks good.
     
  18. Nov 28, 2013 #17
    Thank you for the support and patience!
     
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