How to Find Distance with Coordinates?

Finding the distance between two points in a plane is a fundamental concept in geometry. This concept can be extended to three-dimensional space as well. Let’s dive into how to calculate the distance between two points using their coordinates.

Distance Formula in Two Dimensions

If you have two points, say
$A(x_1, y_1)$ and $B(x_2, y_2)$, the distance between them can be found using the distance formula:

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

Example

Suppose you have points $A(3, 4)$ and $B(7, 1)$. Plugging these values into the distance formula:

$d = sqrt{(7 – 3)^2 + (1 – 4)^2}$

$d = sqrt{4^2 + (-3)^2}$

$d = sqrt{16 + 9}$

$d = sqrt{25}$

$d = 5$

So, the distance between points $A(3, 4)$ and $B(7, 1)$ is 5 units.

Distance Formula in Three Dimensions

When dealing with three-dimensional space, the distance formula extends to include the z-coordinates. For points
$A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, the distance is given by:

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

Example

Consider points $A(1, 2, 3)$ and $B(4, 6, 8)$. Using the three-dimensional distance formula:

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

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

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

$d = sqrt{50}$

$d = 5sqrt{2}$

So, the distance between points $A(1, 2, 3)$ and $B(4, 6, 8)$ is $5sqrt{2}$ units.

Why It Works

The distance formula is derived from the Pythagorean theorem. In two dimensions, the line segment between two points forms the hypotenuse of a right triangle, with the differences in the x-coordinates and y-coordinates forming the other two sides. In three dimensions, the same principle applies, but with an added z-coordinate difference.

Applications

Knowing how to find the distance between points is useful in many fields such as:

• Geometry: Understanding shapes, sizes, and properties of figures.
• Physics: Calculating the displacement between two points in space.
• Computer Graphics: Determining distances between pixels or objects in a virtual space.
• Navigation: Finding the shortest path between two locations.

Conclusion

The distance formula is a powerful tool in geometry and other fields. Whether you are working in two or three dimensions, understanding how to calculate the distance between points helps you solve a wide range of practical problems.

Citations

1. 1. Khan Academy – Distance Formula
2. 2. Math is Fun – Distance Between 2 Points
3. 3. Purplemath – Distance Formula