A vector is a container where order matters and repititions are allowed. An N-vector has n components (elements), each component called
$U_1, U_2, ... , U_n$
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 Pythagoras’ 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: $\vec(u), \vec(x) \in V \Rightarrow \vec{u} + \vec{x} \in V$

Given any number, R, from the original space (real numbers, integers, etc) and $\vec{u} \in V$ then $\vec{u} * R \in V$

The superscript number ontop of a set of numbers such as $R^n$ 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>.