Step By Step Calculus » 2.6  The Rectangular Coordinate System
Synopsis
 A plane refers to any flat (two dimensional) surface.
 A (two dimensional) coordinate system is a method of assigning location to point on a plane.
 The rectangular coordinate system in the plane is also called the Cartesian coordinate system in the plane.
 This coordinate system is defined by a point called the origin and two perpendicular lines crossing at the origin O O, called the x xaxis and the y yaxis.
 The location of a point P P on the plane is defined by a unique pair of numbers (a,b) (a,b), where a a, b b are the signed (or directed) distances of P P from the y y and x xaxis respectively. We often relabel this point P(a,b) P(a,b) or just (a,b) (a,b) when the location of this point is to be emphasized.
 The origin O O is the unique point located at \left(0,0\right) \left(0,0\right).
 The set of possible locations in the rectangular coordinate system is \mathbb{R}^{2} \mathbb{R}^{2} since each location is equated with two numbers, the former giving the distance along the x xaxis and the later giving the distance along the y yaxis.

The distance d d between two points P_0(x_0,y_0) P_0(x_0,y_0) and P_1(x_1,y_1) P_1(x_1,y_1) in the plane is
\displaystyle \textbf{(Distance Formula)} \qquad d(P_0,P_1)= \, P_0\, P_1 \, =\sqrt{(x_1x_0)^2+(y_1y_0)^2} \, .\displaystyle \textbf{(Distance Formula)} \qquad d(P_0,P_1)= \, P_0\, P_1 \, =\sqrt{(x_1x_0)^2+(y_1y_0)^2} \, .
 For example, the set of points in the plane which lie on or within a circle centered at the origin with radius 33 is given by the set of locations \{P:PO\leq 3\}= \{(x,y)\;\; x^2+y^2\leq 9,\;\;x,y\in\mathbb{R} \} \{P:PO\leq 3\}= \{(x,y)\;\; x^2+y^2\leq 9,\;\;x,y\in\mathbb{R} \} .

The midpoint between the points P_0(x_0,y_0) P_0(x_0,y_0) and P_1(x_1,y_1) P_1(x_1,y_1) is
\displaystyle \textbf{(Midpoint Formula)} \qquad P_{\frac12}\left(x_{\frac12}, y_{\frac12}\right)=\left(\frac{x_0+x_1}{2},\frac{y_0+y_1}{2}\right).\displaystyle \textbf{(Midpoint Formula)} \qquad P_{\frac12}\left(x_{\frac12}, y_{\frac12}\right)=\left(\frac{x_0+x_1}{2},\frac{y_0+y_1}{2}\right).

The point P_pP_p which lies at the fraction pp of the full distance from P_0(x_0,y_0)P_0(x_0,y_0) to point P_1(x_1,y_1)P_1(x_1,y_1) is
\displaystyle \textbf{($p$point Formula)} \qquad P_p=\left((1p)x_0+px_1, (1p)y_0+py_1\right).\displaystyle \textbf{($p$point Formula)} \qquad P_p=\left((1p)x_0+px_1, (1p)y_0+py_1\right).
 The point (x,0)(x,0) is the projection of the point (x,y)(x,y) on the xxaxis and the point (0,y)(0,y) is the projection of the point (x,y)(x,y) on the yyaxis.