Calculating the Distance Between Two Points in a Cartesian Coordinate System

Introduction to Distance in a Cartesian Coordinate System

The concept of distance in a Cartesian coordinate system is fundamental in many areas of mathematics and physics. In this article, we will explore how to calculate the distance between two points in three-dimensional space using the distance formula. This topic is particularly relevant for SEO as it involves mathematical concepts that are often discussed in educational and scientific contexts.

Understanding the Distance Formula in Three Dimensions

The distance formula in three-dimensional space is used to determine the distance between two points (x1, y1, z1) and (x2, y2, z2). The formula is given by:

d √[(x2 - x1)2 (y2 - y1)2 (z2 - z1)2]

Applying the Distance Formula

Let's consider the problem of finding the distance between two points with coordinates (1068, 1279, 89) and (1279, 1068, 98).

Solving the Problem Step-by-Step

Step 1: Identify the Coordinates

Let's denote the first point as P1 (1068, 1279, 89) and the second point as P2 (1279, 1068, 98).

Step 2: Apply the Distance Formula

d √[(1279 - 1068)2 (1068 - 1279)2 (98 - 89)2]

d √[(211)2 (-211)2 (9)2]

d √[2112 2112 92]

d √[44521 44521 81]

d √89123

d ≈ 298.53

Additional Examples for Practice

Now that you understand how to use the distance formula, let's look at a couple of additional examples:

Example 1: Distance Between (1, 2, 3) and (4, 5, 6)

d √[(4 - 1)2 (5 - 2)2 (6 - 3)2]

d √[32 32 32]

d √[9 9 9]

d √27

d ≈ 5.196

Example 2: Distance Between (0, 0, 0) and (3, 4, 5)

d √[(3 - 0)2 (4 - 0)2 (5 - 0)2]

d √[32 42 52]

d √[9 16 25]

d √50

d ≈ 7.071

Conclusion

In conclusion, the distance formula is a powerful tool for calculating the distance between two points in three-dimensional space. It is widely used in various fields such as engineering, physics, and data science. By practicing with different examples, you can gain a better understanding of this concept and improve your problem-solving skills.

For further resources and a deeper exploration of the topic, we recommend exploring educational websites, online courses, and books on coordinate geometry and three-dimensional mathematics. Utilizing these resources can help you enhance your knowledge and application of the distance formula in real-world scenarios.