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Math Help - Struggle with coordinate system

  1. #1
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    Struggle with coordinate system

    Hi,

    I want to determine the unit vectors e1,e2 and e3 i a 3D space.

    e1 = (e1x,e1y,e1z)
    e2 = (e2x,e2y,e2z)
    e3 = (e3x,e3y,e3z)

    e1x, e1y, e1z and e2z is known.

    I dont know if it makes any difference but e3z will always be positive so e3 might be written as e3=(sx,sy,1)*t.

    e1, e2, and e3 shall have length 1 and be perpendicular to each other so the following equations holds,

    e3 = CrossProduct(e1,e2)
    e2 = CrossProduct(e3,e1)
    DotProduct(e1,e2) = 0
    DotProduct(e1,e3) = 0
    DotProduct(e2,e3) = 0


    Anyone know how I could solve this?
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  2. #2
    MHF Contributor

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    Re: Struggle with coordinate system

    You know all three components of e1 and you know e2z so e1.e2= e1xe2x+ e1ye2y+ e1ze2z= 0 is an equation with two unknown values, e2x and e2y. Similarly, e1.e3= e1xe3x+ e1ye3y+ e1ze3z= 0 is an equation with three unknown values, e3x, e3y, and e3z. The requirement that the vectors have unit length gives two more equations. So far that is 4 equations for the 5 unknown values. The requirement that e1Xe2= e3 gives the fifth equation.
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  3. #3
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    Re: Struggle with coordinate system

    Thank you for fast answer. That solution will work but the equations to solve get a bit messy.
    I wonder if there might be a "trick" to solve this problem in an easier way.
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