Mathematics - Vector Question with Solution | TestHub

Planes $P_1,P_2$ have equations $x+y-2z=1$ and $\alpha x-y+z=2$, where $\alpha \in \mathbb{Q}$. Points $P(\alpha ,1,3)$ and $Q(5/3,1,\alpha )$ lie on a plane $P_3$ through the line $P_1\cap P_2$. If $\ell ,m,n$ are direction cosines of this line, find $\left|\frac{\sqrt{35}(\ell +m+n)}{\alpha }\right|$.