Lines in the Plane

A real number plane (Fig. 3) is determined by two perpendicular number lines, called the coordinate axes, which intersect at a point, called the origin of the plane or simply the origin. Each point $P$ in the plane can be described by an ordered pair $(x,y)$ of numbers which specify how far, and in which directions, we must move from the origin to reach the point $P$. The point $P = (x,y)$ can then be located in the plane by starting at the origin and moving $x$ units horizontally and then y units vertically. Similarly, each point in the plane can be labeled with the ordered pair $(x,y)$ which directs us how to reach that point from the origin. In this book, a point in the plane will be labeled either with a name, say $P$, or with an ordered pair $(x,y)$, or with both $P = (x,y)$. This coordinate system is called the rectangular coordinate system or the Cartesian coordinate system after Rene Descartes, and the resulting plane is called the Cartesian Plane.

The coordinate axes divide the plane into four quadrants which are labeled quadrants I, II, III and IV as in Fig. 4 We will often call the horizontal axes the x-axis and the vertical axis the y-axis and then refer to the plane as the xy-plane. This choice of $x$ and y as labels for the axes is simply a common choice, and we will sometimes prefer to use different labels and even different units of measure on the two axes.