1. What is a Cartesian Plane?
The Cartesian Plane is a flat two-dimensional surface where we can locate any point using a pair of numbers called coordinates. These coordinates help us show the exact position of a point.
The system was developed by René Descartes, which is why it is called the Cartesian plane.
1.1. Definition
A Cartesian plane is formed by two perpendicular number lines (axes) that intersect at a point called the origin. Any point on the plane is represented as an ordered pair \((x, y)\).
1.2. Uses of the Plane
We use the Cartesian plane to draw graphs, represent geometric shapes, study algebraic equations, and understand relationships between variables.
2. Axes, Origin and Coordinates
The Cartesian plane consists of two main lines: the x-axis (horizontal) and the y-axis (vertical). They intersect at the origin \((0,0)\).
2.1. x-axis
The x-axis is the horizontal line. Positive values lie to the right of the origin and negative values lie to the left.
2.2. y-axis
The y-axis is the vertical line. Positive values lie above the origin and negative values lie below.
2.3. Origin
The point \((0,0)\) is called the origin. It is the reference point from which distances along the axes are measured.
2.4. Coordinates
Every point on the plane is represented as an ordered pair \((x, y)\), where:
- x = horizontal distance from the origin
- y = vertical distance from the origin
2.4.1. Example
For the point \((3, -2)\):
- Move 3 units right on the x-axis
- Move 2 units down on the y-axis
So the point lies in the fourth quadrant.
3. Quadrants of the Cartesian Plane
The two axes divide the plane into four regions called quadrants. These are numbered in the anticlockwise direction starting from the top-right region.
3.1. Description of the Four Quadrants
| Quadrant | Coordinate Signs \((x, y)\) |
|---|---|
| Quadrant I | \((+, +)\) |
| Quadrant II | \((-, +)\) |
| Quadrant III | \((-, -)\) |
| Quadrant IV | \((+, -)\) |
3.2. Axis Points (Special Case)
If a point lies on an axis, one of its coordinates becomes zero. Such points do not belong to any quadrant.
3.2.1. Example
Point \((0, 5)\) lies on the y-axis since the x-coordinate is zero.
4. Understanding Signs and Movement
To locate a point, we follow a fixed order: first the x-coordinate (left or right), then the y-coordinate (up or down).
4.1. Positive and Negative Directions
- Positive x → right
- Negative x → left
- Positive y → up
- Negative y → down
4.2. Example Points
Consider these points:
- \((4, 5)\) → Quadrant I
- \((-3, 2)\) → Quadrant II
- \((-2, -4)\) → Quadrant III
- \((6, -1)\) → Quadrant IV