Gauss Lemma problem from do Carmo's book

Question

I'm reading DoCarmo's book Riemannian Geometry and in the proof of Gauss lemma at the page 69 he says that it is clear that (2) is satified for $w=w_T$. My question is why? I leave a screenshot of the lemma below.

Answer 1

We have $$|(d\exp_p)_v(v)|=|v|.$$ (This is true because the path $t\mapsto\exp_p((1+t)v)$ is a geodesic wherever defined, and the magnitude of the velocity of a geodesic is constant in time (as the Levi-Civita connection is compatible with the metric)). Now, by construction, we have $w_t=av$ for some $a\in\mathbb{R}$. Hence $$\begin{align}\langle(d\exp_p)_v(v),(d\exp_p)_v(w_t)\rangle&=a|(d\exp_p)_v(v)|^2\\&=a|v|^2\\&=\langle v,w_t\rangle.\end{align}$$