Answer
They are neither orthogonal nor parallel.
Work Step by Step
The vectors ${u}=(-\sin \theta, \cos \theta, 1), \quad {v}=(\sin \theta,-\cos \theta, 0)$ are parallel. Since $$u\cdot v=-\sin^2 \theta -\cos^2 \theta +0=-1.$$ then they are neither orthogonal nor parallel.
