3.2 Cartesian System

Description

Quick Overview

The Cartesian system introduces a method for locating points in a plane using two perpendicular axes, the x-axis and y-axis, and is foundational for coordinate geometry.

Standard

This section explains the Cartesian coordinate system, where points are identified based on their distances from two perpendicular axes. It introduces concepts like coordinates, quadrants, and the significance of the origin, providing a basis for understanding concepts in coordinate geometry.

Detailed

Cartesian System

The Cartesian system is a way of representing points in a two-dimensional space using two perpendicular lines, called the x-axis and y-axis. This section begins with a brief overview of how points can be defined in a linear system and transitions into the more complex idea of a two-dimensional coordinate plane.

Key Points:

  1. Definition of Axes: The x-axis is horizontal, while the y-axis is vertical. Their intersection, known as the origin, is denoted as O.
  2. Coordinates: Any point on the plane can be represented as coordinates (x, y), where 'x' denotes the distance from the y-axis (horizontal axis) and 'y' denotes the distance from the x-axis (vertical axis).
  3. Quadrants: The plane is divided into four quadrants based on the signs of the coordinates:
  4. 1st Quadrant: (+, +)
  5. 2nd Quadrant: (−, +)
  6. 3rd Quadrant: (−, −)
  7. 4th Quadrant: (+, −)
  8. Significance of Coordinates: Different combinations of x and y values not only determine a point's location but also indicate its quadrant. Additionally, the specific order of coordinates is crucial; (x, y) is different from (y, x).
  9. Origin: The origin (0, 0) acts as the reference point for all other coordinates.

The system allows for a clear and precise representation of spatial relationships, making it fundamental for higher mathematical concepts.

Key Concepts

  • Coordinate System: A method for pinpointing the location of a point in a plane using two axes.

  • Quadrants: The four sections of the Cartesian plane formed by the intersecting axes.

  • Origin: The (0, 0) point where the x and y axes intersect.

  • Coordinates: The (x, y) values that specify a point's position relative to the origin.

Memory Aids

🎵 Rhymes Time

  • The origin's point is zero and zero, where all lines meet, that's where we go!

📖 Fascinating Stories

  • Imagine a treasure map where the x-axis is a horizontal river and the y-axis is a vertical mountain. By using coordinates, you can find treasure by identifying its exact spot.

🧠 Other Memory Gems

  • Remember 'OX is positive and OY is upward' to recall the direction of positive coordinates.

🎯 Super Acronyms

Think 'Q1, Q2, Q3, Q4' to remember the four quadrants and their signs

  • (+
  • +)
  • (−
  • +)
  • (−
  • −)
  • (+
  • −).

Examples

  • The coordinates of a point located four units to the right and three units up from the origin is (4, 3).

  • A point at (-2, -5) indicates it is two units left and five units down from the origin.

Glossary of Terms

  • Term: Cartesian Plane

    Definition:

    A two-dimensional plane defined by a horizontal line (x-axis) and a vertical line (y-axis) intersecting at the origin.

  • Term: Coordinates

    Definition:

    A pair of values (x, y) that define the position of a point in the Cartesian plane.

  • Term: Origin

    Definition:

    The intersection point of the x-axis and y-axis, represented as (0, 0).

  • Term: Quadrants

    Definition:

    The four sections of the Cartesian plane created by the axes.

  • Term: Abscissa

    Definition:

    The x-coordinate of a point in the Cartesian plane.

  • Term: Ordinate

    Definition:

    The y-coordinate of a point in the Cartesian plane.