In general,, 在,著(another way of writing 着) and 正在 are different when they are used to modify a verb as a tense marker in Mandarin Chinese. When followed by a verb, 正在 and 在 are almost the same. Both of them serve as a tense marker of continuous tense. However, when followed by a place, 在 can only be a preposition to indicate places while 正在 remain to be a tense marker. For example, 这位母亲正在厨房里坐着 and 这位母亲在厨房里坐着 both make sense to me, but they differ in meanings. The former means that this mother is/was sitting in the kitchen, but the latter means that this mother sits/sat in the kitchen. "著" refers to an ongoing state of doing something, which differs from 正在(continuous tense). For instance, 这个男人穿着一顶帽子 To put it more accurately, it means this man is/was in the state of wearing a hat. 这个男人(正)在穿一頂帽子 it means this man is/was putting on a hat. You may see the difference by contrasting the two sentences above. Btw, we don't say 这个男人穿在一頂帽子 cus 在 here precedes a noun, which makes it a preposition and therefore makes no sense to me.(given that a hat is not one of the places for the verb"wear" taking place)

正在的意思就是表示当前的动作，有点类似于英语的doing sth,在的用法就有点像介词（preposition），着的用法有点像语气词（Modal）