Find the projection of M(-1,0,-1) onto the plane 2x+y-z+7=0

Find the projection of $M(-1,0,-1)$ onto the plane $2x+y-z+7=0$.

Step 1: Solution

The perpendicular through $M$ is $x=-1+2t,\ y=t,\ z=-1-t$. Substituting gives $6t+6=0$, so $t=-1$.

\[M'(-3,-1,0)\]

Check: $-6-1+7=0$, and $\overrightarrow{MM'}=\{-2,-1,1\}\parallel\vec n$. The distance is $|MM'|=\sqrt6=\tfrac{6}{\sqrt6}$.