Peak refers to the highest possible level or degree attainable. This could refer to physical height, as would be the peak of Vancouver. It could also refer to the peak of something more abstract, like artistic development. You could also use peak as a verb: The value of Company XYZ stock peaked early last year.
Peek, on the other hand, refers to taking a quick glance or a secret look at something. You could say that Jimmy jumped as high as he could so that he could peek over the fence at his neighbors. It could be considered cheating if you try to peek at your opponent’s cards in poker.
Pique, used in the most common context, is derived from a French word meaning “prick.” That is to say that it stimulates. If I were to say that the Olympus micro four-thirds camera piqued my interest, I would mean that it caught my attention. A pique, when used as a noun, could also mean a sudden outburst of anger, though this usage is much less common.
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