# Is average of two random directions also a random direction?

Given two uniformly random directions on a hemisphere, n0 and n1, is the normalized sum of these vectors also a uniformly random direction on the same hemisphere?

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No - average directions away from the edge are more likely than average directions near the edge of the hemisphere

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@Henry gave you a good answer, I just wanted to supplement it with some visuals. Using normalized standard Gaussian 3D vector to produce points, uniformly distributed on $S^2$ (see near the end of this article on MathWorld) in Mathematica:

The above visualizes distribution for the normalized sum of $n=1$, $n=2$, $n=4$ and $n=8$ vectors uniformly distributed on a hemisphere. Computation of the probability for the $z$-component of such a random point to be above $1/2$ then follows and shows greater concentration of points near then north pole as $n$ increases.

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Here is a suggestion: add a graphics for the sum of, say, 4 or 5 uniform unit vectors, the concentration towards the pole should be more visible. –  Did May 24 '12 at 18:52
@Dider Thanks for the suggestion. I have just updated the graphics. –  Sasha May 24 '12 at 19:03
Magnifique. :-) –  Did May 24 '12 at 19:07