Write in the form vec r·vec N=α the equation of the plane…

Write in the form $\vec r\cdot\vec N=\alpha$ the equation of the plane parallel to $\vec r\cdot(2\vec i-\vec j)=5$ and passing through $M_1(0,1,2)$.

Step 1: Solution

Take $\vec N=2\vec i-\vec j$ and $\alpha=\vec r_1\cdot\vec N=2\cdot0-1\cdot1+0\cdot2=-1$.

\[\vec r\cdot(2\vec i-\vec j)=-1\qquad(2x-y+1=0)\]

Both planes are parallel to $Oz$, since $N_z=0$. Their distance is $\tfrac{|5-(-1)|}{\sqrt5}=\tfrac{6}{\sqrt5}$.