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Complex Vectors

  1. Jul 11, 2009 #1
    1. The problem statement, all variables and given/known data
    A complex vector is written as,
    [tex]\hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y} = \hat{x} + j\hat{y}[/tex],

    where [tex]\omega[/tex] is the angular velocity, and the vector rotates counterclockwise in the x-y plane.


    If a unit vector is rotated in the x-y plane but is phase shifted by 45degrees, then:

    [tex]\hat{v}(t) = (\frac{1}{\sqrt{2}} + j\frac{1}{\sqrt{2}})\hat{x} + \frac{1}{\sqrt{2}} - j\frac{1}{\sqrt{2}})\hat{x} \Rightarrow (cos(\omega t + 45^{\circ})\hat{x} + (cos(\omega t - 45^{\circ})\hat{y}[/tex]

    Can someone explain to me why there are terms [tex]\frac{1}{\sqrt{2}}[/tex] in the equation above. I always thought a 45 degree triangle had sides of [tex]\sqrt{2}, \sqrt{2}, 2[/tex], but not sure how the coefficient [tex]\frac{1}{\sqrt{2}}[/tex] is obtained.


    thanks,


    JL
     
  2. jcsd
  3. Jul 11, 2009 #2

    tiny-tim

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    Homework Helper

    Hi JL! :smile:

    (you've got your x's and y'x mixed up … and you can get LaTeX to write big brackets "to fit" by typing \left( and \right) :wink:)

    Because cos45º = sin45º = 1/√2 (a 45 degree triangle also has sides of 1, 1/√2, 1/√2) :smile:
     
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