December 30, 2022

Distance Between Two Points - Formula, Derivation, Examples

The theory of distance is critical in both math and daily life. From straightforward calculating the length of a line to designing the shortest route among two points, understanding the length between two points is crucial.

In this blog article, we will explore the formula for distance within two points, work on some examples, and discuss real-life utilizations of this formula.

The Formula for Distance Within Two Points

The distance among two points, usually denoted as d, is the extent of the line segment linking the two extremities.

In math, this can be represented by drawing a right triangle and utilizing the Pythagorean theorem. Per the Pythagorean theorem, the square of the length of the extensive side (the hypotenuse) is equal to the sum of the squares of the lengths of the two other sides.

The formula for the Pythagorean theorem is a2 + b2 = c2. As a consequence, √c2 will equal the length, d.

In instance of finding the distance between two locations, we could represent the points as coordinates on a coordinate plane. Let's say we have point A with coordinates (x1, y1) and point B at (x2, y2).

We could thereafter employ the Pythagorean theorem to derive the following formula for distance:

d = √((x2 - x1)2 + (y2 - y1)2)

In this formula, (x2 - x1) portrays the distance on the x-axis, and (y2 - y1) portrays the distance along y-axis, constructing a right angle. By considering the square root of the sum of their squares, we get the distance between the two points.

Here is a visual depiction:

Instances of Applications of the Distance Formula

Now that we have the formula for distance, let's look at few examples of how it can be used.

Finding the Distance Within Two Points on a Coordinate Plane

Imagine we have two points on a coordinate plane, A with coordinates (3, 4) and B with coordinates (6, 8). We will utilize the distance formula to figure out the distance between these two locations as ensues:

d = √((6 - 3)2+ (8 - 4)2)

d = √(32 + 42)

d = √(9 + 16)

d = √(25)

d = 5

Consequently, the span within points A and B is 5 units.

Calculating the Length Among Two Locations on a Map

In addition to finding length on a coordinate plane, we could further utilize the distance formula to calculate distances within two points on a map. For example, assume we have a map of a city with a scale of 1 inch = 10 miles.

To figure out the length between two points on the map, similar to the airport and the city hall, we can easily measure the distance within the two locations employing a ruler and change the measurement to miles using the map's scale.

When we calculate the length among these two locations on the map, we figure out it is 2 inches. We change this to miles using the map's scale and find that the real length within the airport and the city hall is 20 miles.

Working out the Length Within Two Locations in Three-Dimensional Space

In addition to finding distances in two dimensions, we can also use the distance formula to figure out the distance between two points in a three-dimensional space. For instance, assume we have two points, A and B, in a three-dimensional space, with coordinates (x1, y1, z1) and (x2, y2, z2), respectively.

We will utilize the distance formula to figure out the distance between these two points as follows:

d = √((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2)

Using this formula, we can calculate the length within any two locations in three-dimensional space. For example, if we have two locations A and B with coordinates (1, 2, 3) and (4, 5, 6), individually, we could find the length between them as ensues:

d = √((4 - 1)2 + (5 - 2)2 + (6 - 3)2)

d = √(32 + 32 + 32)

d = √(9 + 9 + 9)

d = √(27)

d = 3.16227766

Hence, the length within points A and B is roughly 3.16 units.

Uses of the Distance Formula

Now that we have seen some examples of using the distance formula, let's explore some of its uses in mathematics and other fields.

Calculating Length in Geometry

In geometry, the distance formula is utilized to measure the length of line segments and the sides of triangles. For instance, in a triangle with vertices at points A, B, and C, we utilize the distance formula to calculate the lengths of the sides AB, BC, and AC. These lengths could be used to measure other characteristics of the triangle, for example its area, perimeter and its interior angles.

Solving Problems in Physics

The distance formula is also used in physics to solve questions comprising distance, speed, and acceleration. For instance, if we recognize the initial location and velocity of an object, also the time it takes for the object to transport a specific length, we can utilize the distance formula to calculate the object's concluding position and speed.

Analyzing Data in Statistics

In statistics, the length formula is frequently used to calculate the length within data points in a dataset. This is beneficial for clustering algorithms, that segregate data points that are close to each other, and for dimensionality reduction techniques, which depicts high-dimensional data in a lower-dimensional space.

Go the Distance with Grade Potential

The distance formula is an essential idea in math which allows us to work out the between two location on a plane or in a three-dimensional space. By utilizing the Pythagorean theorem, we could obtain the distance formula and apply it to a magnitude of scenarios, from calculating distances on a coordinate plane to analyzing data in statistics.

Understanding the distance formula and its uses are essential for anyone interested in math and its uses in other areas. If you're having difficulties with the distance formula or any other math concept, reach out to Grade Potential tutoring for personalized help. Our expert instructors will assist you master any mathematics topic, from algebra to calculus and beyond.

Back to Blog