Distance between Two Points in a Plane

Distance between Two Points in a Plane:

Cartesian Coordinate System:

  • Concept: The distance between two points in a plane can be calculated using the coordinates of those points.
  • Formula: The distance formula between two points (x1,y1) and (x2,y2) is given by: d=(x2x1)2+(y2y1)2

Steps to Calculate Distance:

  1. Identify Coordinates: Determine the coordinates of the two points given.
  2. Apply Formula: Plug the coordinates into the distance formula.
  3. Calculate: Use arithmetic operations (addition, subtraction, and squaring) as per the formula.
  4. Square Root: Calculate the square root of the sum of squares to find the distance.

Properties:

  • Non-Negative Value: The distance between two points is always non-negative.
  • Symmetry: The distance between point A and B is the same as between point B and A.
  • Triangle Inequality Theorem: For any three points, the sum of the distances between any two points is greater than the distance between the remaining two points.

Applications:

  • Geometry: Measuring lengths of line segments, diagonals of shapes.
  • Physics: Calculating distances between objects or points in space.
  • Engineering: Designing layouts, structures, and calculating dimensions.

Practical Examples:

Example 1: Distance between Two Cities:

  • Coordinates: Represent cities as points on a map using latitude and longitude coordinates.
  • Distance Calculation: Use formulas such as Haversine or Vincenty to find the distance between cities on the Earth's surface.

Example 2: Distance in 3D Space:

  • Coordinates: Use three-dimensional coordinates (x, y, z) to calculate distances between points in three dimensions.
  • Distance Formula: Extends to 3D space using d=(x2x1)2+(y2y1)2+(z2z1)2

Importance:

  • Navigation: Finding distances between locations on maps and GPS systems.
  • Mathematical Analysis: Critical in geometry, trigonometry, and calculus.
  • Real-World Applications: Used extensively in science, engineering, and various fields for measurement and calculations.

Example:

Consider two points, A and B, with coordinates A(3, 5) and B(-2, -1). Find the distance between these two points.

Steps to Solve:

  1. Identify Coordinates:

    • Point A: x1=3,y1=5
    • Point B: x2=2,y2=1
  2. Apply Distance Formula: d=(x2x1)2+(y2y1)2

  3. Substitute Values: d=(23)2+(15)2

  4. Perform Arithmetic: d=(5)2+(6)2 d=25+36 d=61

  • Final Answer: The distance between points A and B is 61 units.

Interpretation:

The distance between points A and B in the given coordinate system is approximately 7.81 units, calculated using the distance formula. This distance represents the length of the straight line connecting these two points on the plane.