Is the statement, "If you extend your hand to something that is not yours, I will cut off your hand," considered a vow, and what is the correct formula for a vow?

A vow (Nadhir) is the commitment of a legally responsible Muslim (mukallaf) to an act of devotion to Allah. It can be made verbally or through a substitute such as writing or a gesture. A vow is valid with any utterance that indicates commitment, such as "It is Allah's due upon me that I sacrifice a camel." As for the statement you mentioned, it is not a vow; rather, it is a threat and is not permissible for raising children.