I am reading Lee's book "Riemannian manifolds: An Introduction To Curvature". Let $(M,g)$ be a Riemannian manifold. Fix $p\in M.$ It is well known that the exponential map $\exp_p$ map is a local diffeomorphism at $0.$ We can use the identification $T_pM$ to be $\mathbb R^n$ by fixing an orthonormal frame of $T_pM.$ Then we can use the exponential map to define a smooth coordinate chart on a neighbourhood of $p\in M.$ A coordinate chart obtained in this fashion is called a normal coordinate chart. Let $(\mathcal U,x^i)$ be a normal coordinate system. Exercise 5.6 of Lee asks to prove that ``At any point $q\in\mathcal U -p,$ $\partial/\partial r$ is he velocity vector of he unit speed geodesic from $p$ to $q$, and therefore has unit length with respect to $g.$"
Lee defines $r(x):=(\sum_i{x^i}^2)^{\frac{1}{2}}$ and $\partial/\partial r:=\frac{x^i}{r}\partial/\partial x^i$. I can show the following. Let $V:=\exp_p^{-1}(q).$ Then consider the maximal geodesic $\gamma_V$ passing through $p$ and initial velocity $V.$ We know that in normal coordinates $\gamma_V(t)=(tV^1,\dots,tV^n)$. Therefore, after easy calculation $\partial/\partial r|_{\gamma_V(t)}=|V|^{-1}\gamma_V(t)$ and if $|V|=1$ probably we have what Lee asks as $\gamma_V$ is also constant speed. But how can we guarantee that there exists $V$ with $|V|=1.$ This can be shown if there exists a $|V|>1$. Then we can scale it by using star-convexity. But what if $|V|<1?$ Also what about uniqueness Lee is mentioning?