I wanted to understand the question here a little better: Let $r = b - Ax$ be the corresponding residual vector. Which of the following three vectors is a possible value for $r$? Why?
If I understand correctly, the hint posted by the one answerer there implies that I should find the range of $A$ and then check whether the vectors from (a), (b), and (c) are orthogonal to see if they are truly residuals in this case. Lets call these vectors $r_1, r_2, r_3$ respectively.
I know the range of $A$ can be defined as the span of its columns. My question here is, should I start finding all possible dot products of $r_1, r_2, r_3$ with the columns of $A$, and check if any of them come out to zero? I'm not sure that would work. How do I check if $r_1, r_2, r_3$ is orthogonal to the range?
While it's true that the residual vector needs to be orthogonal to every linear combination of the columns (i.e. every vector in the columnspace), you don't need to compute the dot product with every such vector. It suffices to just check that the possible residual vector is orthogonal with a basis (or even just a spanning set) for the columnspace.
Since every column is in the columnspace, certainly every residual vector must be perpendicular to every column. Conversely, if $r$ is perpendicular to every column, then by linearity of the map $x \mapsto x \cdot r$, then $x$ is perpendicular to every linear combination of columns.
So, you really don't have to worry about checking every vector in the columnspace.