2. You should have learned that the scalar product of 2 vectors equals zero if the 2 vectors are perpendicular to each other. Take the ribs of the umbrella's roof as a vector pointing at a point of the roof. This point has the coordinates P(x, y, z) that means we are talking about an arbitrary point of the plane. The fixed point is the point of intersection between the handle and the roof.
Then the vector must lie completely in the plane.
3. The normal vector of a plane is . According to 2.) the product of must equal zero:
This equation is valid for every point in the plane thus this equation describes the plane completely.
Sorry if these "explanations" are too simple for you.