My math book gives this theorem:
Let $f$ be a rigid motion in a real inner product space $X$, and let $T (x) := f (x) - f (0)$. Then T is an orthogonal transformation.
From this theorem, we are asked to prove a lemma.
Part of this lemma is: $(T(x),T(y))=(x,y)$
My thought was $(T(x),T(y))=(f(x)-f(0),f(y)-f(0))=(x-0,y-0)=(x,y)$, as desired.
Is this enough of a proof? My book gives a different proof using $||T (x)-T(y)||^2$, but I do not really follow that proof and to me what I did seems to make sense. Am I correct in that assumption?