Why are the coefficients of the equation of a plane the normal vector of a plane?
I borrowed the below picture from Pauls Online Calculus 3 notes:
And I think the explanation he provides is great, however, I don't understand how one of the concepts work.
If the equation of a plane is $ax+by+cz=d$ how is it that $\overrightarrow n = \langle a,b,c \rangle$? From the picture below I suppose I can see this since $\overrightarrow r_0$, if continued past the plane, would clearly be perpendicular, but what about $\overrightarrow r$? That one is clearly not perpendicular if extended past the plane?
Sorry if what I'm asking is confusing.