yeah but does it imply it needs to be orthogonal because it is symmetric and The problem's statement of "3x3 symmetric matrix" combined with asking for a "third linearly independent eigenvector" leverages the powerful property of symmetric matrices: they are always orthogonally diagonalizable. This means the most natural and direct way to find a third linearly independent eigenvector is to find one that is orthogonal to the given two, as this will ensure it's from a different eigenspace (if eigenvalues are distinct) or completes an orthogonal basis within an eigenspace (if eigenvalues are repeated). By finding a vector orthogonal to both v 1 and v 2 , we guarantee it's linearly independent from them and fulfills the requirements of being an eigenvector for a symmetric matrix