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Linear algebra

  1. Dec 21, 2003 #1
    Find 2 vectors u and v such that they are perpendicular one of the vector is twice the magnitude of the other. And the sum vector of u and v is [6,8]

    I did:

    Let u=[a,b]
    Let v=[c,d]
    Let |u|=2|v

    u.v=ac+bd=0

    |u+v|=|u|^2 + |v|^2

    But |u|=2|v|

    |u+v|=5|v|^2

    5|v|^2=100

    |v|^2=20

    Im stuck after this
     
  2. jcsd
  3. Dec 21, 2003 #2

    Hurkyl

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    You forgot to write ^2 on |u+v|

    Anyways, can you write |v|^2 in terms of a, b, c, and d? Do you know what |u|^2 is? Have you used the fact that u + v = [6, 8] yet?
     
  4. Dec 21, 2003 #3
    So

    |u|^2+|v|^2=|u+v|^2

    But since |u|^2=4|v|^2

    5|v|^2=36+64
    |v|^2=20

    now where can I go?
     
  5. Dec 21, 2003 #4

    Hurkyl

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    Can you write |v|^2 in terms of a, b, c, and d? Do you know what |u|^2 is? Have you used the fact that u + v = [6, 8] yet?
     
  6. Dec 21, 2003 #5
    |v|^2 = c^2+d^2

    |v|^2 = 1/2 (a^2 + b^2)


    1/2(a^2+ b^2)= c^2 + d^2

    Is that what you mean?
     
  7. Dec 21, 2003 #6

    Hurkyl

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    Can you think of anything better you can do with those first two equatinos?
     
  8. Dec 21, 2003 #7
    Since |v|^2=2|u|^2,

    |v|^2 = c^2+d^2

    |u|^2 = 1/2 (c^2 + d^2)


    |u|^2 + |v|^2 = 100

    3/2 (c^2 + d^2) = 100

    I dont know where Im headed
     
  9. Dec 21, 2003 #8

    Hurkyl

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    what formulas do you have involving |v|^2?
     
  10. Dec 21, 2003 #9
    Projection of u on v

    u on v = [u.v/|v|^2] |v|
     
  11. Dec 21, 2003 #10

    Hurkyl

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    I guess I should have asked this first off...

    You know that your goal is to find 4 equations involving only a, b, c, d which you can solve, right?
     
  12. Dec 21, 2003 #11
    4 equations?? I didnt know that...I've been focusing on finding a and b
     
  13. Dec 21, 2003 #12

    Hurkyl

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    Well, you have 4 unknowns; a, b, c, and d. In general, you need 4 equations to solve for all 4 of them.

    Sometimes you can get lucky and you can find two equations that involve only a and b, but in general that won't happen (and I'm pretty sure it doesn't here)...


    You've already found one good equation:

    ac + bd = 0


    You just need 3 more! You can get 2 more equations out of what you've told me about |v|^2...


    Oh, BTW, if |u| = 2|v|, then |u|^2 = 4|v|^2
     
  14. Dec 21, 2003 #13
    Ill try this and post a little later, thanks man
     
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