When you peek at something, you look at it quickly. It’s usually done through some opening, for example, a keyhole, and it’s done quickly, because you are doing it secretly.
- Susan peeked in through the keyhole, but couldn’t see anything.
- My neighbour always peeks out at me from behind the curtains when I get home late.
When you take a look at something, you turn your attention to it and examine it. It’s not done quickly.
- Can you take a look at my essay before I hand it in?
- The doctor took a look at my throat.
The expression take a peek is an informal way to say take a quick look:
- Shall we take a peek at the new bookstore in Oxford street?
- I took a peek outside, but it was raining so I stayed at home.
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