Let a unit vector $\overrightarrow{OP}$ make angles $\alpha,\beta,\gamma$ with the positive directions of the coordinate axes $OX, OY, OZ$ respectively, where $\beta\in\left(0,\tfrac{\pi}{2}\right)$. If $\overrightarrow{OP}$ is perpendicular to the plane through points $(1,2,3)$, $(2,3,4)$ and $(1,5,7)$, then which one of the following is true?