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The tangent plane at the point x=a, y=b is the plane that contains all the tangent vectors (at the point) of curves on the surface that pass through the point (a,b,f(a,b)). Another way to define it is to say that this is the plane through the point (a,b,f(a,b)) which is perpendicular to the NORMAL to the surface at that point. The NORMAL to the surface at a point is just defined as the direction perpendicular to all tangent vectors to the surface at that point. Therefore all we need to do to find the equation of the tangent plane is to find a normal to the surface at that point.
The plane spanned by the three points , , and on a curve as . Let be a point on the osculating plane, then. where denotes the scalar triple product. The osculating plane passes through the tangent. The intersection of the osculating plane with the normal plane is known as the (principal) normal vector. The vectors and (tangent vector and normal vector) span the osculating plane. From MathWorld--A Wolfram Web Resource.