I was confused about what the dot product represents and I think I have grasped that now from this post What does the dot product of two vectors represent? .
However, I still cannot understand what $r·n = a·n$ represents in the same way. I am not sure if this is the universal notation for it, but I was told that '$r$' is any point on a plane $(x,y,z)$ and '$n$' is the normal to the plane with '$a$' being any known point on the plane. I can't understand why the dot product of the normal and ($O$ - any point on the plane) would be the same as the dot product of the normal and ($O$ - one particular point on the plane) as ($O$ - a point on the plane) would be a different vector every time where n stays constant surely resulting in a different dot product.
Probably haven't articulated my misunderstanding too well but any help appreciated.