# Produce an perpendicular vectors that follows a specific distribution

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.

Thank you.

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?

• I don't think you're looking for this type of answer but a proof and I can't do that but you can simply take and create a matrix of normally distributed vectors then use the QR decomposition to produce $Q$ now take $q_{1}$ and perform a test to determine if the orthogonal vectors are normally distributed.. i did it and they are. – воитель Aug 9 '18 at 17:49
• @RHowe thanks for the comment. $Q$ is an orthogonal matrix indeed, but what I want is to obtain a Gaussian distributed random vector that it orthogonal to an arbitrary vector which I do not know its distribution. – darkmoor Aug 9 '18 at 18:17