First consider vectors of unit length, say on the unit sphere. Now I want to give some magnitude to these vectors and I want the magnitude to be chosen from the normal distribution. In one dimension that is easy:
However, if my vector is arbitrarily pointed I cannot take from this distribution since I cannot have a negative magnitude.
The only way I can think of giving the vector a normal distribution of lengths is by using
which forces the randomly chosen quantity to be > 0. Does this seem like the proper procedure?