In word2vec, the task is to learn to predict which words are most likely to be near each other in some long corpus of text. For each word $c$ in the corpus, the model outputs the probability distribution $P(O=o|C=c)$ of how likely each other word $o$ in the vocabulary is to be within a certain number of words away from $c$. We call $c$ the "center word" and $o$ the "outside word".
We choose the softmax distribution as the output of our model: $$P(O=o|C=c) = \frac{\exp(\textbf{u}_{0}^{T} \textbf{v}_{c})}{\sum_{w \in \text{Vocab}} \exp(\textbf{u}_{w}^{T} \textbf{v}_c)}$$
where $\textbf{u}_0$ and $\textbf{v}_c$ are vectors that represent the outside and center words respectively.
Question. What do the vectors $\textbf{u}_0$ and $\textbf{v}_c$ look like? Are they just one-hot-encodings? Do we need to learn them too? Why is this useful?