I'm getting really confused about when can I use

and the following identity, which I'm not sure if it's vectorial (did I write it correctly?)

I'd be really happy to see a formal proof of how those are developed, and to understand in which case do I use each of them. unfortunately I wasn't able find anything (probably I wasn't searching right, but I'm not sure what else to write other than "velocity of rotating object proof")

I'm getting really confused about when can I use

$$ \vec{v} = \dot{r}\hat{r} + \omega \times r \hat{\theta} $$

and the following identity, which I'm not sure if it's vectorial (did I write it correctly?)

$$ \vec{v} = \omega r \hat{\theta} $$

I'd be really happy to see a formal proof of how those are developed, and to understand in which case do I use each of them. unfortunately I wasn't able find anything (probably I wasn't searching right, but I'm not sure what else to write other than "velocity of rotating object proof")

I'm getting really confused about when can I use

and the following identity, which I'm not sure if it's vectorial (did I write it correctly?)

I'd be really happy to see a formal proof of how those are developed, and to understand in which case do I use each of them. unfortunately I wasn't able find anything (probably I wasn't searching right, but I'm not sure what else to write other than "velocity of rotating object proof")

thank you very much for your time and attention!

I'm getting really confused about when can I use

and the following identity, which I'm not sure if it's vectorial (did I write it correctly?)

I'd be really happy to see a formal proof of how those are developed, and to understand in which case do I use each of them. unfortunately I wasn't able find anything (probably I wasn't searching right, but I'm not sure what else to write other than "velocity of rotating object proof")