I dug this problem up from an old exam. I am not asking how to solve them, but I want to get a "feel" for the problem. I could technically solve this brutally,I just want to develop some problem solving skills.
For (a), I notice that the first two vectors must be part of the basis. The only problem are the last two vectors. Is there a way (without computing) to determine whether one of them belongs to the basis or not? I do not want check whether one of them is a linear combination of the other or not.
(b) misread the problem. edit: looks another row-reduction problem...but correct me if I am wrong.
(c) Looks like another annoying row-reduction problem.
Here were my initial thoughts on how to solve each part.
Hint for a:
Find the reduced form of the matrix whose columns are the given vectors, and then take those vectors corresponding to columns with leading ones.
Hint for b:
If two vectors are orthogonal, their dot product is $0$.
Hint for c:
Express the given vector as a linear combination of the vector given in part a. For the three known components, you should obtain a system of equations (three unknowns and three equations). It seems like there is a unique solution.