I am interested to produce a random Gaussian distributed vector perpendicular to a single or a collection of orthogonal predefined vectors. In the simple case of a single predefined vector, my first thought was to compute a Gaussian random vector, compute the projection over the initial vector and then compute the perpendicular one. However, I am suspecting that the produced perpendicular vector does not follow a Gaussian distribution or any other distribution we are interested, e.g half norm distribution. Can you please help me to prove my suspicion? Also, is it possible to produce new random vectors that are perpendicular to a single or multiple orthogonal predefined vectors?
Any help is highly appreciated.
EDIT: I was wondering, if it is possible to rotate a random vector to make it perpendicular to a predefined one and remain Gaussian? Is a distribution rotational invariant?