Let S be the subspace of R4 containing all vectors with x1 + x2 + x2 + x4 = 0. Find a basis for the space S orthogonal, containing all vectors orthogonal to S.
There are a few ways to solve this but first note that basis for S is
So you could just find a vector perpendicular to these three above vectors.
Or if you take the gradient of you get
you can check using the vectors above that this is orthogonal to the space S.