You did not mention one important point: di+ ej+ fk is anormalvector, perpendicular to the plane. Now, if r is a vector lying in the plane, in particular if r= <x- a, y- b, z- a> where (a, b, c) is given as a specific point in the plane and (x, y, z) represents a variable point in the plane, then r is perpendicular to di+ ej+ fk and so their dot product is 0.

If you have three points in the plane, say, (a, b, c), (d, e, f), (g, h, i), then (a- d, b- e, c- f) and (a- g, b- h, c- i) are vectors lying in the plane and so you can take di+ ej+ fk to be the cross product of those two vectors.