A vector is a container where order matters and repititions are allowed.
An N-vector has n components (elements), each component called

Two vectors can be added like so:

You can only add two vectors if the vectors have the same number of components.

Scalar multiplcation is where you increase every item in a vector by R. Let R be a real number then:

The length of a vector a, |a|, can be calculated with Pythagorasâ€™ formula

Some important facts:

A vector space is a list, V, of n-vectors where each vector is defined strictly using a type of number (real, rational etc) and if and only if:

Given any number, R, from the original space (real numbers, integers, etc) and then

The superscript number ontop of a set of numbers such as is the set of all real valued (numbers that are in the real numbers) n-vectors where each component is in the set of Real numbers.

2-vectors have 2 dimensions, X and Y: <X, Y>. 3-vectors have <X, Y, Z>.