Let $A$ be a 3x3 real symmetric matrix, and let $v_1, v_2$ be orthogonal eigenvectors of $A$. Suppose $v_3$ is orthogonal to both $v_1$ and $v_2$. Is $v_3$ necessarily an eigenvector of $A$?
To put it another way: by the spectral theorem we know that $\mathbb{R}^3$ has an orthonormal basis consisting of eigenvectors of $A$, but can we extend any two orthogonal eigenvectors of $A$ into such a basis?