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In this answer to the question "Intuitive explanation for why centripetal acceleration is $\frac{v^2}{r}$": https://physics.stackexchange.com/a/190532/262601 (excerpt from the answer is attached below), I don't understand why it says that "The only difference between the position and velocity is that we rotated by 90 degrees and multiplied the length by v/r". I understand why it needs to be rotated by 90 degrees, but I don't understand how we come up with the v/r.

A point is moving around a circle. It has a blue position vector and a red velocity vector, like this:

enter image description here

The position vector stays the same length and rotates around and around in a circle. Because the position vector is changing, it has a derivative. That derivative is the velocity.

Because we're always going the same speed, the velocity vector also stays the same length. Because the velocity is always 90 degrees rotated from the position, the velocity is also going around in a circle. In other words, the velocity vector is doing exactly the same thing as the position vector is doing; rotating and staying constant length. The only difference between the position and velocity is that we rotated by 90 degrees and multiplied the length by v/r.

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You are looking for a relationship between $v$ and $r$. If you start with $r$, and you want to "go" to $v$, then you multiply by $\dfrac vr$ since $r\cdot\dfrac vr=v$.

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