Finding vectors orthogonal to the span of a matrix

[Fractal20]

Homework Statement

My linear algebra is rusty. So to go from a reduced QR factorization to a complete QR factorization (ie the factorization of an over determined matrix) one has to add m-n additional orthogonal vectors to Q. I am unsure on how to find these.

If it is extending a 3x2 to a 3x3 I know I can take the cross product of the first two columns etc. but I am unsure of a process to do this in a general case.

It seems like one possibility would be to look at the null space of the orthogonal projector onto the range of A (the pseudo inverse is it called)? But that seems time consuming. It there a quick method to find a vector orthogonal to a set of orthonormal vectors? Thanks y'all!***edit
Or, I just realized, guess I could find the nullspace of the transpose of A. Is there a particular approach anybody would suggest?

Homework Equations

The Attempt at a Solution

 

[lippyka]

I think I may have realized the answer. I will still post this in case it helps somebody else. To find the additional orthogonal vectors to extend a reduced QR factorization to a complete one you can take the nullspace of the transpose of A.