Distance formula
Understanding the Distance Formula
The distance formula is a fundamental concept in geometry, allowing us to calculate the distance between two points in a coordinate system. This formula is derived from the Pythagorean theorem and can be applied in various dimensions, including 2D and 3D spaces.
Distance Formula in 2D
In a two-dimensional space, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the following formula:
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
This formula calculates the length of the hypotenuse of a right-angled triangle formed by the two points and the projections onto the x-axis and y-axis.
Example in 2D
Let's calculate the distance between the points $A(3, 4)$ and $B(7, 1)$.
$$ d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $$
The distance between points A and B is 5 units.
Distance Formula in 3D
In three-dimensional space, the distance formula is extended to include the z-coordinate. The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is:
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} $$
Example in 3D
Consider two points $C(2, -1, 3)$ and $D(1, 4, 2)$. The distance between them is:
$$ d = \sqrt{(1 - 2)^2 + (4 - (-1))^2 + (2 - 3)^2} = \sqrt{(-1)^2 + 5^2 + (-1)^2} = \sqrt{1 + 25 + 1} = \sqrt{27} $$
The distance between points C and D is $\sqrt{27}$ units.
Differences and Important Points
Here's a table summarizing the differences and important points of the distance formula in 2D and 3D:
| Aspect | 2D Distance Formula | 3D Distance Formula |
|---|---|---|
| Formula | $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ | $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ |
| Coordinates Used | $(x_1, y_1)$ and $(x_2, y_2)$ | $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ |
| Dimensions | 2D (plane) | 3D (space) |
| Derived From | Pythagorean Theorem | Pythagorean Theorem in 3D |
| Use Cases | Planar geometry, maps, graphs | Spatial geometry, physics, 3D modeling |
Applications
The distance formula is widely used in various fields such as:
- Mathematics: In geometry to solve problems involving distances and perimeters.
- Physics: To calculate the displacement between two points in space.
- Computer Graphics: To determine the distance between pixels or objects in 3D modeling.
- Navigation: To find the shortest path between two locations on a map.
Conclusion
The distance formula is a powerful tool for calculating the straight-line distance between two points in both 2D and 3D spaces. Understanding this formula is essential for students and professionals who deal with geometric calculations and spatial analysis. By applying the formula correctly, one can solve a wide range of problems in mathematics, physics, and other related disciplines.