Radial and directional parts of a random vector

Let X be a random vector on and let R = [short parallel]X[short parallel] and for R [not equal to] 0 let W = W/R. Necessary and sufficient conditions are given for R and W to be independent. If X has a non-singular normal distribution we show that the following three conditions are equivalent. 1. (i) the components of X are independent and identically distributed with 0 means and positive variances. 2. (ii) W is uniformly distributed on the unit sphere. 3. (iii) R and W are independent.