December 30, 2022

Distance Between Two Points - Formula, Derivation, Examples

The theory of length is critical in both

pasting

and routine life. From easily measuring the length of a line to calculating the quickest route among two locations, comprehending the distance within two points is vital.

In this blog, we will inspect the formula for distance between two points, review some examples, and discuss realistic uses of this formula.

The Formula for Length Between Two Points

The distance within two points, often signified as d, is the length of the line segment connecting the two points.

Mathematically, this can be represented by drawing a right triangle and employing the Pythagorean theorem. Per the Pythagorean theorem, the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the distances 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 the circumstance of finding the distance within two locations, we could portray the extremities as coordinates on a coordinate plane. Let's assume we have point A with coordinates (x1, y1) and point B at (x2, y2).

We can further use the Pythagorean theorem to acquire the ensuing formula for distance:

d = √((x2 - x1)2 + (y2 - y1)2)

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

Here is a visual depiction:

Instances of Applications of the Distance Formula

Considering we have the formula for distance, let's check out some examples of how it can be utilized.

Calculating the Distance Among the Two Locations on a Coordinate Plane

Assume we have two extremities on a coordinate plane, A with coordinates (3, 4) and B with coordinates (6, 8). We will employ the distance formula to calculate the length within these two locations as follows:

d = √((6 - 3)2+ (8 - 4)2)

d = √(32 + 42)

d = √(9 + 16)

d = √(25)

d = 5

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

Calculating the Distance Within Two Locations on a Map

In addition to working out the length on a coordinate plane, we could further utilize the distance formula to calculate lengths within two locations on a map. For example, assume we have a map of a city along a scale of 1 inch = 10 miles.

To find the distance between two locations on the map, such as the airport and the city hall, we can easily work out the length between the two locations utilizing a ruler and change the measurement to miles utilizing the map's scale.

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

Calculating the Length Between Two Points in Three-Dimensional Space

In addition to calculating distances in two dimensions, we could also utilize the distance formula to figure out the length among two locations 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 use the distance formula to figure out the length between these two locations as follows:

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

Utilizing this formula, we could determine the length between any two points in three-dimensional space. For example, if we possess two points A and B with coordinates (1, 2, 3) and (4, 5, 6), respectively, we can figure out the distance within 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

Thus, the length between locations A and B is just about 3.16 units.

Applications of the Distance Formula

Now once we have seen some instances of utilizing the distance formula, let's study few of its uses in mathematics and other areas.

Measuring Length in Geometry

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

Solving Problems in Physics

The distance formula is further utilized in physics to work out problems involving speed, distance and acceleration. For example, if we perceive the original location and velocity of an object, as well as the time it requires for the object to transport a certain distance, we can use the distance formula to calculate the object's final position and speed.

Analyzing Data in Statistics

In statistics, the distance formula is often utilized to calculate the distances between data points in a dataset. This is useful for clustering algorithms, that segregate data points that are near to each other, and for dimensionality reduction techniques, this portrays high-dimensional data in a lower-dimensional space.

Go the Distance with Grade Potential

The distance formula is ac crucial concept in mathematics which enables us to figure out the length between two location on a plane or in a three-dimensional space. By utilizing the Pythagorean theorem, we can derive the distance formula and implement it to a variety of situations, from calculating length on a coordinate plane to analyzing data in statistics.

Understanding the distance formula and its applications are essential for everyone interested in mathematics and its uses in other fields. If you're having difficulties regarding the distance formula or any other math concept, contact Grade Potential tutoring for personalized help. Our expert instructors will assist you conquer any mathematics topic, from algebra to calculus and beyond.

Back to Blog