Can anyone explain the following representation of the dual space of $\mathbb{S}^2(ie. T^*\mathbb{S}^2)$?
$$T^∗\mathbb{S}^2=\lbrace(u, v) \in \mathbb{R}^3×\mathbb{R}^3:|u|= 1,\ 〈u, v〉= 0\rbrace$$
In a book, they give this representation but for me, this looks like representation for $T\mathbb{S}^2$. Maybe I am wrong. Please explain my mistake.
Thanks, Advance.