$$\Delta = \left| {\matrix{ {\cos (\beta - \alpha )} & {\sin \beta } & 1 \cr { - \sin (\beta - \alpha )} & { - \cos \beta } & 1 \cr {\cos (\beta - \alpha - \theta )} & {\sin (\beta - \theta )} & 1 \cr } } \right|$$

$${R_3} \to R(\cos \theta {R_1} + \sin \theta {R_2})$$

$$ = \left| {\matrix{ {\cos (\beta - \alpha )} & {\sin \beta } & 1 \cr { - \sin (\beta - \alpha )} & { - \cos \beta } & 1 \cr 0 & 0 & {1 - (\cos \theta + \sin \theta )} \cr } } \right|$$

$$ = - [1 - (\cos \theta + \sin \theta )]\cos (2\beta - \alpha ) \ne 0$$

$$ = 2\beta < {\pi \over 2}$$ and $$\alpha > 0$$

$$ \Rightarrow (2\beta - \alpha ) < {\pi \over 2} \Rightarrow \cos (2\beta - \alpha ) \ne 0$$

Also, since $$0 < \theta < {\pi \over 2} \Rightarrow \cos \theta + \sin \theta > 1$$

$$\Delta \ne 0$$

$$\Rightarrow$$ P, Q and R are not collinear.