1.16 Coordinates in a Plane

What we learn here:

 

Click here to read about History of Coordinate System

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Points

In Section 1.3 “Sets of Numbers“, we showed how the real number system can be used to assign coordinates to points on a line.

 

Summary of assigning coordinates to points on a line

 

In the plane, to determine the position of a point $P$, we need two real numbers, namely a pair of numbers $(a,b)$. We draw two perpendicular straight lines, label them as the $x$ and $y$ axes, designate their intersection as the origin, choose the same unit distance on both axes, and lay off distances from the origin along the axes. The distances to the right along the $x$-axis are taken as positive and those to the left as negative. In the same way, distances upward along the $y$-axis are taken as positive and those downward as negative.

In this case, the position of a point $P$ in the plane can be clearly indicated by drawing two straight lines perpendicular to the axes, intersecting the $x$ and $y$ axes at $M$ and $N$, respectively. Two signed distances $a=\overline{OM}$ and $b=\overline{ON}$ are called respectively the abscissa (or the $\mathit{x}$ component) of $P$ and the ordinate (or the $\mathit{y}$ component) of $P$ and together known as the rectangular coordinates or the Cartesian coordinates (in honor of of René Descartes) of the point. In this case, we write $P(a,b)$ (see the following figure).

 

• The $x$ and $y$ axes divide all of the plane into four regions called quadrants. The quadrants are numbered I, II, III, and IV (called first, second, third, and fourth quadrants) as in the above figure.

Because in general $(a,b)\ne (b,a)$, the pair $(a,b)$ is often referred to as an ordered pair.

more explanation about ordered pairs

 

• Instead of “the point with coordinates $(a,b)$,”  we often simply say “the point $(a,b)$.”

• The set of all ordered pairs is denoted by $\mathbb{R}\times \mathbb{R}$ or ${\mathbb{R}}^2$

$${\mathbb{R}}^2=\{(a,b)|a,b\in \mathbb{R}\}.$$

 

Distance between points

Because we used the same unit of length on both axes, we can apply geometry to express the distance between two points in terms of their coordinates. Consider two points $P(x_1,y_1)$ and $Q(x_2,y_2)$ as in the following figure. If we apply the Pythagorean theorem to the right triangle $PRQ$, we realize the distance between $P$ and $Q$, denoted by $d(P,Q)$ or $|PQ|$ is given by:

 

 

The Midpoint Formula

• Remark that $|PM|=|MQ|$ (or $d(P,M)=d(M,Q)$)