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Find |4a - 5b|, |a| = 1, |b|=2, a*b = -(1/3)*|a| |b|

  1. Oct 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Find |4a - 5b|, |a| = 1, |b|=2, a*b = -(1/3)*|a| |b|
    where

    a and b are vectors.


    3. The attempt at a solution

    I use big letters (X,Y) for the vector `a`, while small letters
    for the vertor `b` (x,y).

    (data) =>
    (1) X^2 + Y^2 = 1
    (2) x^2 + y^2 = 4
    (3) Xx + Yy = -2/3

    --- Data processing

    (11) X = (1 - Y)^.5
    (22) x = (1 - y)^.5
    (33) (Yy)^.5 + Yy - y^2 - 4y^2 = 0

    Solving (33) gives for Y
    (44) [tex] Y = -.5 \pm .5 * \sqrt ( \frac { 17y^2 -1} {y^2 -1} ) [/tex]

    I am not certain whether "brute force" is the best tool here.
     
  2. jcsd
  3. Oct 15, 2009 #2

    lanedance

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    Homework Helper

    i don't think you have to explictly solve for (x,y) & (X,Y) in fact i doubt you can...

    you eant to find the length of a linear combination of the vectors, which will eb independent fo any ratation to the axis you make, so why not make a = (1,0) then use the dot product to give the angle between a & b to fi9nd b
     
  4. Oct 16, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    [itex]|4a-5b|^2[/itex][itex]= (4a-5b)\cdot(4a- 5b)[/itex][itex]= 16|a|^2- 40a\cdot b+ 25|b|^2[/itex]

    That's pretty much all you need, isn't it?
     
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