Definition:Coordinate System/Coordinate

Elements of Ordered Pair
Let $\left({a, b}\right)$ be an ordered pair.

The following terminology is used:
 * $a$ is called the first coordinate
 * $b$ is called the second coordinate.

This definition is compatible with the equivalent definition in the context of Cartesian coordinates.

Some authors use the terms first component and second component instead.

Coordinate System
Let $\left \langle {a_n} \right \rangle$ be an ordered basis of a unitary $R$-module $G$.

Then $\left \langle {a_n} \right \rangle$ can be referred to as a coordinate system.

Coordinate
Let $\left \langle {a_n} \right \rangle$ be a coordinate system of a unitary $R$-module $G$.

Let $\displaystyle x \in G: x = \sum_{k=1}^n \lambda_k a_k$.

The scalars $\lambda_1, \lambda_2, \ldots, \lambda_n$ can be referred to as the coordinates of $x$ relative to $\left \langle {a_n} \right \rangle$.

Origin
The origin of a coordinate system is the zero vector.

In the $xy$-plane, it is the point $O = \left({0,0}\right)$

Comment
It's an awkward word coordinate. It really needs a hyphen in it to emphasise its pronounciation (loosely and commonly: coe-wordinate), and indeed, some authors spell it co-ordinate. However, this makes it look unwieldy.

An older spelling puts a diaeresis indication symbol on the second "o": coördinate. But this is considered archaic nowadays and few sources still use it.