What is the vector proof of the pythagorean theorem?

I know vector proofs use vector addition, subtraction, and scalar multiplication, but I can't find anything to help me prove this theorem. Please help in detail.|||The dot product of vector [a,b] with itself gives its squared length, and since [a,b] dot [a,b] = (a)(a) + (b)(b) = a^2 + b^2, that is the squared length of that vector. But that vector is just the hypotenuse of the the right triangle with base the vector [a,0] and altitude the vector [0,b]. Consider the points (0,0), (a,0) and (a,b) to see what I am talking about. This shows that the hypotenuse of a right triangle with legs a and b has squared length a^2 + b^2, which is the Pythagorean theorem.





However, I am quite sure that there is circular reasoning going on in this "proof".