Finding eigenvector only knowing others eigenvectors.

Since $A$ is symmetric, the eigenvectors (for distinct eigenvalues) are orthogonal.

So, find which of the vectors is orthogonal to the first two.

(1,1,-3) is.

Hint: the condition $A^t = A$ allows you to use the spectral theorem.

Hint: Specifically, the spectral theorem implies there is an orthonormal basis of eigenvectors of $A$.