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

1. Oct 5, 2011

### karkas

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. Oct 5, 2011

### Dick

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?

3. Oct 5, 2011

### karkas

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).

4. Oct 5, 2011

### SammyS

Staff Emeritus
y has cosine like the others, not sine.

5. Oct 5, 2011

### Dick

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?

6. Oct 5, 2011

### karkas

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.

7. Oct 5, 2011

### Dick

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.

8. Oct 5, 2011

### Dick

Oh, that was you SammyS. Want to take it from here??

9. Oct 5, 2011

### karkas

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?

10. Oct 5, 2011

Yes.