$x,y$ are perpendicular if and only if $x\cdot y=0$. Now, $||x+y||^2=(x+y)\cdot (x+y)=(x\cdot x)+(x\cdot y)+(y\cdot x)+(y\cdot y)$. The middle two terms are zero if and only if $x,y$ are perpendicular. So, $||x+y||^2=(x\cdot x)+(y\cdot y)=||x||^2+||y||^2$ if and only if $x,y$ are perpendicular. ( I copied this)
I think this argument is circular because the property
$x\cdot y=0 $ implies $x$ and $y$ are perpendicular
comes from the Pythagorean theorem.
Oh, it just came to mind that the property could be derived from the law of cosines. The law of cosines can be proved without the Pythagorean theorem, right, so the proof isn't circular?
Another question: If the property comes from the Pythagorean theorem or cosine law, then how does the dot product give a condition for orthogonality for higher dimensions?
Edit: The following quote by Poincare hepled me regarding the question:
Mathematics is the art of giving the same name to different things.
Best Answer
I think the question mixes two quite different concepts together: proof and motivation.
The motivation for defining the inner product, orthogonality, and length of vectors in $\mathbb R^n$ in the "usual" way (that is, $\langle x,y\rangle = x_1y_1 + x_2y_2 + \cdots + x_ny_n$) is presumably at least in part that by doing this we will be able to establish a property of $\mathbb R_n$ corresponding to the familiar Pythagorean Theorem from synthetic plane geometry. The motivation is, indeed, circular in that we get the Pythagorean Theorem as one of the results of something we set up because we wanted the Pythagorean Theorem.
But that's how many axiomatic systems are born. Someone wants to be able to work with mathematical objects in a certain way, so they come up with axioms and definitions that provide mathematical objects they can work with the way they wanted to. I would be surprised to learn that the classical axioms of Euclidean geometry (from which the original Pythagorean Theorem derives) were not created for the reason that they produced the kind of geometry that Euclid's contemporaries wanted to work with.
Proof, on the other hand, consists of starting with a given set of axioms and definitions (with emphasis on the word "given," that is, they have no prior basis other than that we want to believe them), and showing that a certain result necessarily follows from those axioms and definitions without relying on any other facts that did not derive from those axioms and definitions. In the proof of the "Pythagorean Theorem" in $\mathbb R^n,$ after the point at which the axioms were given, did any step of the proof rely on anything other than the stated axioms and definitions?
The answer to that question would depend on how the axioms were stated. If there is an axiom that says $x$ and $y$ are orthogonal if $\langle x,y\rangle = 0,$ then this fact does not logically "come from" the Pythagorean Theorem; it comes from the axioms.