Exercise 6.1.10

Two vectors are orthogonal means that the inner product of them is $0$. So we have

To deduce the Pythagorean Theorem in $R^2$, we may begin by a right triangle $\mathit{ABC}$ with the right angle $B$. Assume that $x=\mathit{AB}$ and $y=\mathit{BC}$. Thus we know the length of two leg is $\parallel x\parallel$ and $\parallel y\parallel$. Finally we know $\mathit{AC}=x+y$ and so the length of the hypotenuse is $\parallel x+y\parallel$. Apply the proven equality and get the desired result.