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An isosceles triangle is formed by a unit vector in the x-direction and another in a random direction. Find the distribution of the length of the third side in three dimensions.

I know how to do this problem in two dimensions. The angle varies with uniform distribution, so after some trigonometry and calculation, you get (2/pi)arcsin(x/2). In three dimensions, the angle that varies is the angle between the random vector and the x-y plane. Then I'm stuck as to how the angle relates to the length of the third side (if at all).

Thanks so much!

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    Is the vector in the random direction a unit vector? If so, your distribution will depend on two angles...2011-04-26

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It might be useful to know the differential for area in spherical coordinates: $d\Omega = \sin(\theta)\;d\theta\;d\phi$ And recall that the total surface area of a sphere is $4\pi$.