Concept Notes: The Cartesian Plane, Axes, Origin & Quadrant Signs
1. Foundational Architecture of the Cartesian Plane
The Cartesian coordinate system provides a one-to-one correspondence between every point in a 2D plane and an ordered pair of real numbers $(x, y) \in \mathbb{R}^2$.
Figure 1.1: The Cartesian Plane — Horizontal X-Axis, Vertical Y-Axis, Origin (0,0) and the Four Quadrant Signs
- Axes of Reference:
- X-Axis ($X'OX$): Horizontal number line. Right of $O$ is positive; left is negative.
- Y-Axis ($Y'OY$): Vertical number line. Above $O$ is positive; below is negative.
- The Origin ($O$): The unique point of intersection $(0, 0)$.
- Ordered Pair Components:
- Abscissa ($x$): Perpendicular distance of point from the $y$-axis.
- Ordinate ($y$): Perpendicular distance of point from the $x$-axis.
- Point notation: $\mathbf{P(x, y)}$. Note that if $x \neq y$, then $(x, y) \neq (y, x)$.
2. Sign Conventions Across Quadrants
| Quadrant | Sign of Abscissa ($x$) | Sign of Ordinate ($y$) | Coordinate Nature | Example Point |
|---|---|---|---|---|
| Quadrant I | Positive ($x > 0$) | Positive ($y > 0$) | $(+, +)$ | $A(3, 7)$ |
| Quadrant II | Negative ($x < 0$) | Positive ($y > 0$) | $(-, +)$ | $B(-5, 4)$ |
| Quadrant III | Negative ($x < 0$) | Negative ($y < 0$) | $(-, -)$ | $C(-2, -6)$ |
| Quadrant IV | Positive ($x > 0$) | Negative ($y < 0$) | $(+, -)$ | $D(4, -8)$ |
3. Points on the Coordinate Axes
- On the X-Axis: Every point on the $x$-axis has $y = 0$. Form: $\mathbf{(x, 0)}$. Example: $(5, 0), (-3, 0)$.
- On the Y-Axis: Every point on the $y$-axis has $x = 0$. Form: $\mathbf{(0, y)}$. Example: $(0, 6), (0, -4)$.
- Origin ($O$): Lies simultaneously on both axes: $\mathbf{(0, 0)}$.
- Points on the axes do not belong to any of the four quadrants.