 Solution

The dot product is ${\mathbf{u}}_{+}\cdot {\mathbf{u}}_{-}=-\cos \theta \sin \theta +\sin \theta \cos \theta =0$. Each planar vector is perpendicular to ${\mathbf{u}}_3$ because its third component vanishes. Each has norm one, so the three vectors form an orthonormal eigenbasis. This also follows from the spectral theorem for the real symmetric coefficient matrix.