Say I have a set of vectors and I want to find a basis for the orthogonal complement. I know a way to do this is to just calculate the dot product of each given vector with a general vector. This way you get a system of linear equations and you solve it (this is just finding the nullspace of the matrix that represents the system).
But I came across another way of finding the complement. From what I understood, you can extend the given set to a basis for the entire space (e.g. extend a set of 3 vectors in R⁴ to a basis of R⁴) and then use the Gram-Schmidt process on this extended basis. Apparently you have to use the vector you added to extend the basis last (which I believe I understand why). In the case of R⁴, the last vector is a basis for the orthogonal complement.
The question is: is this method valid? I can see why this process would work, but I don't understand why you would use it.