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!