Three Words, Fifty Thousand Slots

A tokenizer with a vocabulary of $V = 50{,}000$ has handed out these IDs:

Each token is then one-hot encoded: a vector of length $50{,}000$ that is $0$ everywhere except for a single $1$ at the position given by its ID. Call these vectors $\mathbf{c}_{\text{cat}}$, $\mathbf{c}_{\text{car}}$ and $\mathbf{c}_{\text{dog}}$.

The straight-line (Euclidean) distance between two vectors is

$\lVert \mathbf{u} - \mathbf{v} \rVert = \sqrt{\sum_{i=1}^{V} (u_i - v_i)^2}$

What are the two distances $\lVert \mathbf{c}_{\text{cat}} - \mathbf{c}_{\text{car}} \rVert$ and $\lVert \mathbf{c}_{\text{cat}} - \mathbf{c}_{\text{dog}} \rVert$?