Eigenvalues are of course scaling factors by which a transformation may multiply a corresponding eigenvector. So in your case, you know that
if , given and , then and behave as .
You now know there exists a vector such that . Thus is an eigenvector with eigenvalue . Since this does not match any of the previous eigenvalues, you know that this new is linearly independent of and (explain why). Then, you should be able to find the components of just as you found .