1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cos^2(φ_1) +cos^2 (φ_2) + cos^2(φ_3) = 1 in a three dimensional cartesian system

  1. Oct 5, 2011 #1
    1. The problem statement, all variables and given/known data
    I seem to be stuck for an assignment that I have for one of my classes, in which we are asked to prove that cos^2(φ_1) +cos^2 (φ_2) + cos^2(φ_3) = 1 in a three dimensional cartesian system, where φ_1 ,φ_2, φ_3 are the angles that a random vector r (x,y,z) is to the x,y and z axxi respectively.

    2. Relevant equations
    Prove that cos^2(φ_1) +cos^2 (φ_2) + cos^2(φ_3) = 1.


    3. The attempt at a solution
    I have made various attempts at linking the angles together and forming some kind of equation but none of them lead to the solution. It just seems really random to me, maybe I'm wrong because it's so early in the morning...
     
  2. jcsd
  3. Oct 5, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If phi_1 is the angle (x,y,x) makes with the x axis, then x=sqrt(x^2+y^2+z^2)*cos(phi_1), yes? That's just trig. What are the other two coordinates?
     
  4. Oct 5, 2011 #3
    I'm guessing you mean r(x,y,z). So its gotta be y=sqrt(x^2+y^2+z^2)*sin(phi_2) and z=sqrt(x^2+y^2+z^2)*cos(phi_3).
     
  5. Oct 5, 2011 #4

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    y has cosine like the others, not sine.
     
  6. Oct 5, 2011 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Well, I meant (x,y,z) to be the coordinates of the point. Why did you put sin in the y coordinate? sqrt(x^2+y^2+z^2) is the length of the vector. cos is the ratio between the hypotenuse and the coordinate, yeah?
     
  7. Oct 5, 2011 #6
    Oh yeah my bad. I'm then guessing that it's wrong to say that cos(phi_2)=sin(phi_1).I'm not entirely sure which angles our teacher wanted us to use, therefore I'm confused. The fact is that I have formed these equations, but messing with them led me to the beggining which generally means I'm missing something.
     
  8. Oct 5, 2011 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Ah, ok. Then do you understand why those things are true? If so, then compute x^2+y^2+z^2 using x=cos(phi_1)*sqrt(x^2+y^2+z^2), etc.
     
  9. Oct 5, 2011 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Oh, that was you SammyS. Want to take it from here??
     
  10. Oct 5, 2011 #9
    Ah I think I got it and I guess it's just a matter of not spotting the answer, my everlasting doom.


    x^2+y^2+z^2 = |z|^2
    and
    x^2+y^2+z^2 = [ cos^2(phi_1) + cos^2(phi_2) + cos^2(phi_3) ] * ( x^2+y^2+z^2)
    = [ cos^2(phi_1) + cos^2(phi_2) + cos^2(phi_3) ] * |z|^2,
    therefore we have proven it?
     
  11. Oct 5, 2011 #10

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Cos^2(φ_1) +cos^2 (φ_2) + cos^2(φ_3) = 1 in a three dimensional cartesian system
  1. Cos^2(pi/4) help (Replies: 3)

Loading...