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Linear Algebra - Use angles between vectors to find other vectors

  1. Feb 2, 2012 #1
    1. The problem statement, all variables and given/known data

    Find all the unit vectors X element of R^3 that make an angle of pi/4 radians with vector Y = (1,0,1) and an angle of pi/3 radians with vector Z = (0,1,0)


    2. Relevant equations

    For any two vectors X and Y element of R^n, the dot-prodict of X and Y is equals to the length of X times the length of Y times the cosine of the angle between X and Y. That is:

    X*Y = cos(t)|x||y|

    3. The attempt at a solution

    let X = ([itex]x_{1}[/itex],[itex]x_{2}[/itex],[itex]x_{3}[/itex])

    We need X*Y = cos([itex]\pi[/itex]/4) |x| |y|
    and X*Z = cos([itex]\pi[/itex]/3)|x||z|

    We want the length of X, that is |x|, to be 1 so I'll assume that it is 1 for now (this could be a bad idea).

    X*Y= [itex]x_{1}[/itex] + [itex]x_{3}[/itex] = srt(2)/2 * sqrt(2) = 1
    X*Z = [itex]x_{2}[/itex] = 1/2 * sqrt(2) = sqrt(2)/2

    Solving for [itex]x_{1}[/itex] and [itex]x_{2}[/itex] in terms of [itex]x_{3}[/itex] we get:

    X = (1, 1/2, 0) + [itex]x_{3}[/itex](-1, 0, 1)

    Problem: X is not a unit vector for all [itex]x_{3}[/itex], so we haven't really found a formula for "all unit vectors" which fulfill the initial requirements. This is where I'm stuck! I thought about including another variable for |x| in the restrictions and adding a further restriction that [itex]x_{1}[/itex]^2 + [itex]x_{2}[/itex]^2 + [itex]x_{3}[/itex]^2 = 1, but I'm not sure how to handle such a nonlinear constraint!
     
  2. jcsd
  3. Feb 3, 2012 #2

    vela

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    Why is there a factor of ##\sqrt{2}## in the equation for ##\vec{x}\cdot\vec{z}##?
     
  4. Feb 3, 2012 #3
    Apologies! That sqrt(2) should be a 1 (the length of Z). I copied that sqrt(2) from the memory of another similar problem I was working on. I'll edit the original post!

    This doesn't, however fundamentally change things. The question still remains.
     
  5. Feb 3, 2012 #4
    Apparently, I can't edit the original post, so I'll re-write it here:


    3. The attempt at a solution

    let X = ([itex]x_{1}[/itex],[itex]x_{2}[/itex],[itex]x_{3}[/itex])

    We need X*Y = cos([itex]\pi[/itex]/4) |x| |y|
    and X*Z = cos([itex]\pi[/itex]/3)|x||z|

    We want the length of X, that is |x|, to be 1 so I'll assume that it is 1 for now (this could be a bad idea).

    X*Y= [itex]x_{1}[/itex] + [itex]x_{3}[/itex] = srt(2)/2 * sqrt(2) = 1
    X*Z = [itex]x_{2}[/itex] = 1/2 * 1 = 1/2

    Solving for [itex]x_{1}[/itex] and [itex]x_{2}[/itex] in terms of [itex]x_{3}[/itex] we get:

    X = (1, 1/2, 0) + [itex]x_{3}[/itex](-1, 0, 1)

    Problem: X is not a unit vector for all [itex]x_{3}[/itex], so we haven't really found a formula for "all unit vectors" which fulfill the initial requirements. This is where I'm stuck! I thought about including another variable for |x| in the restrictions and adding a further restriction that [itex]x_{1}[/itex]^2 + [itex]x_{2}[/itex]^2 + [itex]x_{3}[/itex]^2 = 1, but I'm not sure how to handle such a nonlinear constraint!
     
  6. Feb 3, 2012 #5

    vela

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    You just need to find the value or values of x3 for which X has unit length. Solve the equation X2=1 for x3.
     
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