Quick A: Do not confuse mathematical concepts with C++ terminology.
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You are getting confused because the mathematical concept of a vector can mean a "collection of data" and that is what you were taught int v was. The actual name for that in C++ (and most other languages) is an "array" not a vector.
The libraries referred to in C++ Primer have a class called "vector" which is an implementation of an array. They are similar, but not the same.
I hope that clears that up a bit. You are probably confused because you were taught that int v is a vector, but it is "not really" in C++. It's an array. Use that term to refer to it. If you ever refer to it as a vector, you will confuse others and yourself.