What is $\alpha^{\vee}$ for any root $\alpha$?

Question

Let $\alpha_i, i \in I$ be simple roots in a root system and $\omega_i, i \in I$ the fundamental weights. The fundamental roots are dual to the fundamental coweights and the fundamental weights are dual to the fundamental coroots: $\alpha_i(\omega_j^{\vee}) = \delta_{ij}$, $\omega_i(\alpha_j^{\vee})=\delta_{ij}$. What is $\alpha^{\vee}$ for any root $\alpha$ in the root system? For example, in type $A_2$, what is $(\alpha_1 + \alpha_2)^{\vee}$? Thank you very much.