One-Dimensional Coordinate System
A one-dimensional coordinate system is the simplest form of geometry.
- Concept: Represents points on a number line.
- Origin and Coordinates: The starting point is the origin, with positive values to the right and negative values to the left.
- Distance Between Points: Conceptually understood as the absolute difference between positions.
- Applications: Measuring motion along a straight road, positions in games, or temperature scales.
Two-Dimensional Coordinate System
- Introduction: Represents points in a plane using ordered pairs (x, y).
- Axes and Quadrants: The x-axis (horizontal) and y-axis (vertical) divide the plane into four quadrants.
- Plotting Points: Each point is uniquely identified by coordinates.
- Applications: Navigation maps, location plotting, economic graphs.
Straight Line in Plane
- Concept: A straight line is the shortest distance between two points.
- Slope and Intercept: They describe how a line is inclined and where it crosses axes.
- Parallel and Perpendicular Lines: Relationships are based on slopes and angles.
- Applications: Geometry, physics motion graphs, economics (supply-demand curves).
Circle
- Definition: A circle is the set of all points equidistant from a fixed center.
- Center and Radius: Core properties that define the circle.
- Position of Points: A point can lie inside, on, or outside the circle.
- Applications: Wheels, design structures, planetary orbits.
Matrices and Determinants Formulas | Properties, Operations & Applications
Ellipse
- Concept: Locus of points where the sum of distances from two foci remains constant.
- Shape: Flattened circle with major and minor axes.
- Applications: Planetary orbits, satellites, communication systems.
Hyperbola
- Definition: Locus of points where the difference of distances from two foci remains constant.
- Branches and Asymptotes: Distinct curves spreading outward.
- Applications: Radio navigation, astronomy, satellite tracking.
Parabola
- Concept: Locus of points equidistant from a fixed point and a line.
- Shape: A U-shaped curve.
- Applications: Satellite dishes, projectiles, reflecting surfaces in headlights and antennas.
Three-Dimensional Coordinate System
- Extension: Adds the z-axis to the 2D system.
- Representation of Points: Each point has coordinates (x, y, z).
- Octants: The 3D equivalent of quadrants.
- Applications: Architecture, computer graphics, physics modeling.
Plane
- Concept: A flat surface extending infinitely in 3D.
- Orientation: Defined conceptually by a point and direction.
- Applications: Construction, geometric modeling, design of 3D objects.
Straight Line in Space
- Concept: A line in 3D is defined by a direction and a point.
- Difference from 2D: Extends across x, y, and z axes.
- Applications: Engineering paths, structural designs, 3D trajectories.
Vector Formulas | Coordinates, Addition, Products & Applications
Quadric Surfaces
- Concept: Surfaces defined by second-degree equations.
- Examples: Ellipsoids, paraboloids, hyperboloids.
- Applications: Physics, 3D modeling, computer graphics.
Sphere
- Definition: Set of all points in space equidistant from a center.
- Radius and Center: Fundamental to defining a sphere.
- Applications: Balls, planets, architectural domes.
