在【zài】

在 says where something is. Used on its own it is the whole verb, so no other verb is needed.

在 plus a place, before the verb

When the sentence also has an action, 在 and its place go in front of the verb. English puts them at the end, and that habit produces wǒ我chī fàn吃饭zài在shí táng食堂, which is wrong.

A thing needs a localizer, a place name does not

Words like 北京, 学校 and 家 are places already. Ordinary nouns are not, so 上, 里 or 下 has to be added to turn them into one. zài在zhuō zi桌子 is not a location; zài在zhuō zi桌子shǎng上 is.

Negation

不在 for a general statement, 没在 for a moment that has passed.

Questions

在哪儿 asks where, and the question word sits where the answer would sit.