Say i have an n dimesional Random vector $ \vec{r} $ whose components $ \ \{{x_1 ,x_2....,x_n\}} $ are iid Gaussian Random variables, and I need to show that unit vectors $\hat{n}=\frac{\vec{r}}{|r|}$ are distributed uniformly on unit n sphere.
my approach to this problem was first by finding the joint pdf
$P(x_1,x_2,...x_n)=\frac{1}{(\sqrt{2\pi \sigma^2 })^n}\exp{(\Sigma_{i=1}^n x_i^2/2 \sigma^2)}$
then I think I need to marginalize wrt the angle but don't know how to proceed with it .
