I have seen a proof for the formula of A.B = ||A|| ||B|| cos(theta)[ proof using the diagram and cosine rule]. In the proof they have assumed that distributive property of dot product is right. diagram is given below c.c =(a-b).(a-b) = a^2 +b^2 -2(a.b) [ here they used distributive law] I have seen another proof for the distributive property of dot product. There they have assumed that A.B = ||A|| ||B|| cos(theta),And used projections. They have used the diagram as given below. And projection of vector B on A is ||B||cos(theta) = B.a^ ( a^ is a unit vector in the direction of a vector)[ this is possible if the formula of dot product is assumed to be right. How they can do this , for proving dot product A.B = ||A|| ||B|| cos(theta) they have assumed distributive property to be right and for prooving distributive property they have assumed dot product to be right. Therefore I think that there will be a definition for dot product wether it is A.B = ||A|| ||B|| cos(theta) or A.B = a1a2 +b1b2 +c1c2 (component form). If its a definition then how they have defined it like this.