Determine α,β in α x+β y+2z-1=0 so that the plane is…

Determine $\alpha,\beta$ in $\alpha x+\beta y+2z-1=0$ so that the plane is perpendicular to $\tfrac{x-1}{1}=\tfrac y2=\tfrac{z+1}{-1}$.

Step 1: Solution

The normal must be parallel to $\vec d$: $\tfrac\alpha1=\tfrac\beta2=\tfrac2{-1}$.

\[\alpha=-2,\quad\beta=-4:\qquad -2x-4y+2z-1=0\]

Check: $\vec n=\{-2,-4,2\}=-2\vec d$. The line pierces the plane at $t=-\tfrac5{12}$, i.e. at $\left(\tfrac7{12},-\tfrac56,-\tfrac7{12}\right)$.