A vector is an ordered list of numbers. If the vectors has \(n\) numbers, then it is an \(n\)-dimensional vector. We use \(\overrightarrow{x}\) to represent the vector \(x.\)
For example, suppose you are tracking the amount of apples, oranges, apricots and pears sold at a grocery store. If a customer buys 4 apples, 6 oranges, 0 apricots, and 2 pears, you can represent the information in a vector:
\[<4, 6, 0, 2>\]
This is a 4-dimensional vector.
A vector can be represented as a variable with an arrow. If another customer buys 3 of each fruit, the vector representing what they buy is
\[\overrightarrow{v} = <3, 3, 3, 3>\]
The vector \(<2, 2, 1>\) does not mean anything in this case because it is the wrong dimension. The vector \(<2, 2, 1, 0>\) means 2 apples, 2 oranges, 1 apricot, and 0 pears in this 4-dimensional vector space.
In math, we usually do not assign a meaning to each dimension. We study \(n\)-dimensional vector spaces in general where \(n \geq 2.\)