Intuitively, a plane is a flat surface that extends infinitely in all directions. How can we interpret this definition in a formal way, and in particular use that discussion to derive the standard equation of a plane?
Initially, I had a few different ideas for approaching this, but now it seems like the best is an approach inspired by Calculus III – Equations of Planes (lamar.edu). We will keep deriving intuitively expected conditions from our description “a flat surface that extends infinitely in all directions” until we yield a formal definition.
A condition we can derive from our description (“a flat surface that extends infinitely in all directions”) is: if we pick a point on the plane, then we can find a vector perpendicular to the plane at that point, and furthermore this vector is perpendicular to the plane at any point on the plane (intuitively, “since the plane is flat.”) Furthermore, the set of all such vectors is one-dimensional: 
Note that so far we haven’t formally defined what perpendicularity means here, since it’s not the same as perpendicularity/orthogonality to the vector from the origin to the point. But we will formally interpret this now.
Take any plane and pick a point 
But we can go further: using the fact that two vectors are orthogonal if and only if their dot product is zero, we immediately derive the standard equation form of a plane. Philosophically and from a formalization standpoint, we can either define perpendicularity to a plane as above and then define a plane as the set of all points perpendicular to a given vector, or we can define a plane as any set having the standard equation form.
