It's like a fancy list.

It's a dynamically-sized list of objects of the same type stored contiguously in memory.

dynamically-sized: The size of it can change as needed.

list: It stores multiple things together.

object: A bit of programmer defined data.

of the same type: all the objects in the list are defined the same way

stored contigiously in memory: if you think of memory as a bookshelf then all the objects on the list would be stored right next to each other on the bookshelf rather than spread across the bookshelf.

Many things like each other lined up in a row, and you can take some away or put more in.

It's how you want an array to work.

This is such an understated but useful description in this context. It's also how I understood algebra for applied matrix computation.

Its the algebraic properties that are important, not all vectors are n-tuples, eg the set of polynomials of degree less than n.

You need a basis to coordinate a vector, you can work with vectors without doing that and just deal with the algebraic properties. The coordinate representation is dependent on the basis chosen and isn't fundamental to the vector. So calling them n-tuples isn't technically correct.

You can turn them into a set of coordinates if you have a basis, but the fact that you can do that is because of the algebraic properties so it's those properties which define what a vector is.

No. ArrayList is thread safe and implements the collections API. Vector doesn't. Though if you're using Java, there's almost no instance where you would want to use a Vector instead of ArrayList.