Vector Algebra

Let a unit vector \(\widehat{O P}\) make angles \(\alpha, \beta, \gamma\) with the positive directions of the co-ordinate axes \(\mathrm{OX}\), \(\mathrm{OY}, \mathrm{OZ}\) respectively, where \(\beta \in\left(0, \frac{\pi}{2}\right)\). If \(\widehat{\mathrm{OP}}\) 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?