# 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 $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ is given by: $d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$

#### 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 $d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}+\left({z}_{2}-{z}_{1}{\right)}^{2}}$

### 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: ${x}_{1}=3,{y}_{1}=5$
• Point B: ${x}_{2}=-2,{y}_{2}=-1$
2. Apply Distance Formula: $d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$

3. Substitute Values: $d=\sqrt{\left(-2-3{\right)}^{2}+\left(-1-5{\right)}^{2}}$

4. Perform Arithmetic: $d=\sqrt{\left(-5{\right)}^{2}+\left(-6{\right)}^{2}}$ $d=\sqrt{25+36}$ $d=\sqrt{61}$

• Final Answer: The distance between points A and B is $\sqrt{61}$ 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.