the matrix:

has rank 1, so it has null space of dimension 2. it's clear that (x,y,z) is in this nullspace iff x+y+z = 0, that is if (x,y,z) is of the form: (s,t,-s-t),

which is the plane: s(1,0,-1) + t(0,1,-1). a basis is {(1,0,-1),(0,1,-1)}.

both of your example points lie in this plane. simply take s = c, t = 0, for the first one, and s = -c, t = 2c for the second.