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# 12.1: Coordinates on a Flat Plane

Difficulty Level: At Grade Created by: CK-12

A portrait of Rene Descartes.

A more widely used system are cartesian coordinates, based on a set of axes perpendicular to each other. They are named for Rene Descartes (“Day-cart"), a French scientist and philosopher who back in the 1600s devised a systematic way of labeling each point on a flat plane by a pair of numbers. You may well be already familiar with it.

The system is based on two straight lines (“axes"), perpendicular to each other, each of them marked with the distances from the point where they meet (“origin") --- distances to the right of the origin and above it, the origin being taken as positive and on the other sides as negative.

Graphs use this system, as do some maps.

This works well on a flat sheet of paper, but the real world is 3-dimensional and sometimes it is necessary to label points in 3-dimensional space. The cartesian (x,y)\begin{align*}(x,y)\end{align*} labeling can be extended to 3 dimensions by adding a third coordinate z. If (x,y)\begin{align*}(x,y)\end{align*} is a point on the sheet, then the point (x,y,z)\begin{align*}(x,y,z)\end{align*} in space is reached by moving to (x,y)\begin{align*}(x,y)\end{align*} and then rising a distance z\begin{align*}z\end{align*} above the paper (points below it have negative z\begin{align*}z\end{align*}).

Very simple and clear, once a decision is made on which side of the sheet z\begin{align*}z\end{align*} is positive. By common agreement the positive branches of the (x,y,z)\begin{align*}(x,y,z)\end{align*} axes, in that order, follow the thumb and the first two fingers of the right hand when extended in a way that they make the largest angles with each other.

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