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The following set of rules are called the rules of vector algebra:

addition:

if x=(x₁, x₂, ... , x_n)
and y=(y₁, y₂, ... , y_n),
then x+y=(x₁+y₁, x₂+y₂,... , x_n+y_n);

scalar multiplication:

if x=(x₁, x₂, ... , x_n) is a vector and t is a scalar,
then tx=(tx₁, tx₂, ..., tx_n).

A closed set of objects obeying these rules is a vector space. A single such object
is a vector.