Section 7.1 The Cartesian Plane
Subsection 7.1.1 Coordinates
While most of these modules are focus on the skills and techniques of algebra and functions, this section is devoted to (two-dimensional) geometry. The type of geometry that you need to learn or review here is Cartesian geometry: geometry described by coordintaes.
Coordinates are a way of identifying points in geometry (and, once we have points, also shapes, distances, and many other constructions). The setup of coordinates has several steps. The
- We are working in a two dimensional environment. I'll call this environment simply the plane. If you want to be more specific, you could call is the Cartesian plane to indicate the coordinate system we are using.
- First, we choose a point to act as the centre of the plane. This point is called the origin.
- Going out from the origin are two lines, one horizontal and one vertical. These are called axes. By convention, we usually called the horizontal axis the \(x\)-axis and the vertical axis the \(y\)-axis. The axis have a notion of distance along them, often indicated by marks like on a ruler. Again by convention, we think of moving right as the positive \(x\) direction and moving up as the positive \(y\) direction.
- Once we have the origin and the axis, then any point in the plane can be identified by its position relative to the two axis. The point \((3,6)\) is the unique point that we get to by moving \(3\) units horizontally (to the right) and moving \(6\) units vertically (upward). If we have negative coordinates, we move in the other direction. The point \((-5,-2)\) is the unique point that we get to by moving \(5\) units horizontally (to the left, since it is negative) and \(2\) units vertically (downward, since it is negative).
Figure 7.1.2 shows five points and their coordinates. The point \((0,0)\) is the origin.