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**Laurent** For any differentiable curve $\displaystyle \gamma$ on $\displaystyle M$ such that $\displaystyle \gamma(0)=p$, the function $\displaystyle s\mapsto \|q-\gamma(s)\|^2$ has a minimum at 0, hence its derivative at this point is 0, which gives $\displaystyle 2(q-\gamma(0),-\gamma'(0))=0$, i.e. $\displaystyle (q,\gamma'(0))=0$: $\displaystyle q$ is orthogonal to $\displaystyle \gamma'(0)\in T_pM$. Since $\displaystyle T_pM$ is spanned by the vectors of the form $\displaystyle \gamma'(0)$ (this is even one possible definition), we deduce that $\displaystyle q$ is orthogonal to $\displaystyle T_pM$.