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Finding the a vector using angles

  1. Dec 21, 2011 #1
    I have three non-parallel vectors (whose components are known), v0, v1, v2 and I'd like to find a fourth unit vector, v3, given only the angles between:

    psi, angle between v0 and v3
    theta, angle between v1 and v3
    phi, angle between v2 and v3

    I know that they are related by the dot product:

    v0 . v3 = |v0||v3| cos(psi)
    v1 . v3 = |v1||v3| cos(theta)
    v2 . v3 = |v2||v3| cos(phi)

    this give me a system of equations to solve,

    however, I have a term on the right side |v3|, expands into sqrt(a^2+b^2+c^2) - so its not quite linear.

    If I have the constraint that I want to find a UNIT vector v3, ie. |v3|=1, then can i just set sqrt(a^2+b^2+c^2)=1, to make it a linear system?

    Also - is there an easier way to solve this?
     
    Last edited: Dec 21, 2011
  2. jcsd
  3. Dec 22, 2011 #2
    I guess the equivalent problem would be to solve the following system of equations:

    ax+by+cz=d*sqrt(x^2+y^2+z^2);
    ex+fy+gz=h*sqrt(x^2+y^2+z^2);
    ix+jy+kz=l*sqrt(x^2+y^2+z^2);

    a,b,c,d,e,f,g,h,i,j,k,l constants
    solve for x,y,z
     
  4. Dec 22, 2011 #3

    chiro

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    Science Advisor

    Hey exmachina.

    One thing you may have overlooked: v3 is a unit vector. What does this do to your sqrt(x^2 + y^2 + z^2) term?
     
  5. Dec 23, 2011 #4
    yes I realize that its equal to 1. But when I tried solving some examples setting sqrt(x^2 + y^2 + z^2)=1, the resulting values I get actually wasn't 1.
     
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