Sorry if this is OT, but I'm wondering if there's a specific graphics technique in which it is required to extend a preexisting orthogonal set of vectors (not necessarily to a full basis, but perhaps so). In a matrix theory lecture, the prof said that this computation has a real-world application, but I can't remember him saying what it was. Could be computer graphics, could be a lot of things. I realize that it is only an issue in a high-dimensional space. For example, if you had two orthogonal vectors in 3-space, you could compute their cross product cheaply. I asked at Math SE, and it was not appreciated.
TLDR: Just need the application, if any. You don't have to say how it is computed, but feel free to if you like. I'll read it. I know that orthogonal vectors are often easier to work with than oblique vectors, and other very general math facts like that.