Vector equation of the plane – Normal form

The vector equation of the plane which is at a distance of 'p' from the origin along the unit vector n is r . n = p.

Image

Proof:

Let the perpendicular from the origin 'O' to the plane meet at ' N '.

Then ON = pn

Let P be any point in the plane and OP = r

∴ ON ⊥ NP
⇒ ON . NP = 0
⇒ ON . (OP – ON) = 0
⇒ pn . (r – pn) = 0
⇒ (r . n) – p (n . n) = 0 (∵ r . n = n . r)
⇒ (r . n) – p = 0 (∵ n . n = 1)
⇒ (r . n) = p