Have fun being dead, Gandalfthewhite.
Have you read LotR already? If so, have you read the appendixes, which are found at the end of the book? They should clear out many questions like this. Pretty much is also told about Nümenor, Gondor and Arnor, and Elendil, Isildur and Anarion, so you should be able to find answer to your questions there.
If Arnor's location isn't stated there, then it's, as the Thunder Mistress said, it covered most parts of Eriador. After Earendur(correct me if it was not Earendur) Arnor was divided between his sons. It was divided to three kingdoms, Arthedain, Cardolan and Rhudaur, which are shown on the map. The kingdoms were ruined later though.
(Off-topic, but was the Shire in Arnor? Wasn't Arnor already divided when the hobbits settled in the Shire? It was in Arthedain, wasn't it? Or do I remember wrong?)
Hmm, I wasn't planning to write this much, but at least this should give you an idea of where Arnor was, even though it was already answered :P
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"Out of doubt, out of dark to the day's rising I came singing in the sun, sword unsheathing. To hope's end I rode and to heart's breaking: Now for wrath, now for ruin and a red nightfall!"
~Eomer
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