Let a unit vector OP make angles α, β, γ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where β ∈(0, (π)/2). If OP is…

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