Distance Between Two Points

In geometry, one of the most fundamental and essential concepts is determining the distance between two points. Whether you are solving problems in mathematics, physics, or even in real-life scenarios like navigation, knowing how to find the distance between two points is crucial.

1.0What is Distance?

The distance between two points is simply the length of the straight line that connects them. In simple terms, it's the shortest path between two points, no matter the space in which those points exist. This concept can be applied in both two-dimensional (2D) and three-dimensional (3D) spaces, and the formula to calculate it slightly changes based on the dimensions.

2.0Distance Formula in 2D

Let’s start by considering two points on a two-dimensional plane. The coordinates of these points are:

• $PointA:(x_1,y_1)$
• $PointB:(x_2,y_2)$

The distance formula in a 2D plane is derived from the Pythagorean theorem. According to the theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. To calculate the distance between the two points, we treat the difference in the x-coordinates as one side of a right triangle and the difference in the y-coordinates as the other side.

The formula for distance in 2D is:

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Where:

• d is the distance between the points A and B,
• $(x_1,y_1)and(x_2,y_2)$ are the coordinates of points A and B.

Example:

If you have the points A(1, 2) and B(4, 6), the distance between these two points can be calculated as:

$d=\sqrt{(4-1{)}^2+(6-2{)}^2}$

$d=\sqrt{3^2+4^2}$

$d=\sqrt{9+16}=\sqrt{25}=5$

So, the distance between points A and B is 5 units.

3.0Distance Formula in 3D

In three-dimensional space, we need to account for the additional z-coordinate. If the coordinates of two points in 3D space are:

• $PointA:(x_1,y_1,z_1)$
• $PointB:(x_2,y_2,z_2)$

The formula for the distance between the two points in 3D space becomes:

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

This formula follows the same principle as the 2D distance formula, but we add the difference in the z-coordinates.

Example:

If you have the points A(1, 2, 3) and B(4, 6, 8), the distance can be calculated as:

$d=\sqrt{(4-1{)}^2+(6-2{)}^2+(8-3{)}^2}$

$d=\sqrt{3^2+4^2+5^2}$

$d=\sqrt{9+16+25}=\sqrt{50}\approx 7.07$

So, the distance between points A and B in 3D space is approximately 7.07 units.

4.0Real-Life Applications of Distance Between Two Points

The concept of distance between two points is not just theoretical; it is used in many real-life applications:

1. Navigation and GPS Systems:
   The distance formula is fundamental to navigation systems like GPS. Whether you’re walking, driving, or flying, GPS systems calculate the distance between your location and your destination using similar concepts, though they often consider more factors (like the Earth's curvature in long distances).
2. Physics and Engineering:
   In physics, calculating the distance between two points is critical for determining displacement, velocity, and acceleration. Engineers often use these calculations when designing structures or when calculating forces in mechanical systems.
3. Computer Graphics:
   The distance between two points is essential in 3D modeling and computer graphics. It's used in rendering, creating realistic animations, and even in the collision detection algorithms in video games.
4. Data Science and Machine Learning:
   In data science, the distance between points in multidimensional space (often called Euclidean distance) is used in clustering algorithms, recommendation systems, and classification tasks.