Euclidean Distance Formula Definitions and Examples
In mathematics, the Euclidean distance or Euclidean metric is the “ordinary” straight-line distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. A generalized version of the Euclidean metric is the Minkowski metric. In physics and engineering, the Euclidean distance is used to measure the length of an object or the distance between two points. For example, a carpenter measuring the length of a table would use the Euclidean distance formula. However, if you were to ask a surveyor to measure the same table, they would use the Pythagorean theorem because they are interested in measuring the hypotenuse of the right triangle created by the endpoints of the table.
Euclidean Distance Formula
Assuming that we have two points in some Euclidean space, we can calculate the distance between those points using the Euclidean distance formula. The formula is derived from the Pythagorean theorem and is represented by the following equation:
d = ?((x_2-x_1)^2 + (y_2-y_1)^2)
where d is the distance between the two points, x and y are the coordinates of each point, and 1 and 2 represent the respective points.
The Euclidean distance formula is used in many different fields and applications, including but not limited to: physics, engineering, medicine, biology, robotics, computer vision, pattern recognition, machine learning, artificial intelligence, etc.
One of the most common applications of the Euclidean distance formula is in measuring similarity or dissimilarity between objects. For example, if we have a dataset of images and we want to find out which image is most similar to a given query image, we can use the Euclidean distance formula to calculate the distances between the query image and all other images in the dataset. The image with the smallest distance from the query image is deemed to be most similar to it.
What is Euclidean Distance Formula?
The Euclidean distance formula is a mathematical formula used to calculate the distance between two points in Euclidean space. The formula is derived from the Pythagorean theorem, which states that the square of the hypotenuse (the longest side of a right triangle) is equal to the sum of the squares of the other two sides.
The Euclidean distance between points A and B is given by:
d = ?((x_A-x_B)^2+(y_A-y_B)^2)
where x_A and y_A are the coordinates of point A, and x_B and y_B are the coordinates of point B.
The Euclidean distance formula can be used to calculate distances in any number of dimensions. For example, in three-dimensional space, the formula becomes:
d = ?((x_A-x_B)^2+(y_A-y_B)^2+(z_A-z_B)^2))
Euclidean Distance Formula
The Euclidean distance formula is a mathematical formula used to calculate the distance between two points in Euclidean space. The formula is derived from the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
In Euclidean space, the hypotenuse is the straight line segment that connects two points. The length of this segment can be calculated using the Euclidean distance formula.
The Euclidean distance formula is:
d = ?((x2-x1)^2 + (y2-y1)^2)
Where d is the distance between points x1 and x2, and y1 and y2 are the respective coordinates of each point.
An example of how to use this formula: Suppose we have two points, A and B, with coordinates (4,5) and (-3,1) respectively. To find the distance between these points using the Euclidean distance formula, we would plug these values into the formula as follows:
d = ?((4-(-3))^2 + (5-1)^2)d = ?(7^2 + 4^2)d = ?61d = 7.815…
Euclidean Distance Formula Derivation
The Euclidean distance formula is derived from the Pythagorean theorem. The theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
This can be written as:
$$\text{hypotenuse}^2 = \text{side1}^2 + \text{side2}^2$$
If we take one side to be the difference between two points, then we can write:
$$\text{distance}^2 = \text{difference in x coordinates}^2 + \text{difference in y coordinates}^2$$
Which is the same as:
$$\text{distance} = \sqrt{\text{difference in x coordinates}^2 + \text{difference in y coordinates}^2}$$
Examples Using Euclidean Distance Formula
There are many ways to use the Euclidean Distance Formula. Here are some examples:
Find the distance between two points on a coordinate plane:
For example, find the distance between (-1,2) and (3,4).
First, we need to find the difference between the x-coordinates and the y-coordinates. We do this by subtracting -1 from 3 and 2 from 4. This gives us a difference of 4 for the x-coordinates and 2 for the y-coordinates. Now we can plug these values into the formula:
d = ?(4^2 + 2^2)
d = ?(16 + 4)
d = ?(20)
d = 4.4721…
The Different Types of Euclidean Distance Formulas
There are three different types of Euclidean distance formulas: the straight-line distance formula, the Pythagorean theorem, and the Manhattan distance formula.
The straight-line distance formula is the simplest of the three formulas and is used to calculate the distance between two points that are either horizontally or vertically aligned. To use this formula, simply take the difference between the x-coordinates of the two points and then take the difference between the y-coordinates of the two points. The sum of these two differences will give you the straight-line distance between the two points.
The Pythagorean theorem is a bit more complex than the straight-line distance formula and is used to calculate the distance between two points that are not horizontally or vertically aligned. To use this formula, you must first find the length of each side of the triangle formed by the two points and then use those lengths to solve for the hypotenuse using the Pythagorean theorem. The hypotenuse is always going to be your answer when using this formula.
The Manhattan distance formula is similar to the Pythagorean theorem in that it is used to calculate distances between points that are not horizontally or vertically aligned. However, instead of finding side lengths, you must find horizontal and vertical displacements. These displacements are then added together to give you your final answer, which is always going to be a positive value no matter what your inputs were.
Examples of Euclidean Distance Formula
The Euclidean distance formula is a mathematical formula used to calculate the distance between two points in Euclidean space. The formula is derived from the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
The Euclidean distance between two points, P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ), is given by:
d = ?((x 2 – x 1 )2 + (y 2 – y 1 )2)
For example, if P 1 has coordinates (3, 0) and P 2 has coordinates (0, 4), then the Euclidean distance between these two points is 5.
How to Use the Euclidean Distance Formula
When you’re working with the Euclidean distance formula, there are a few things to keep in mind. First, you need to have at least two points in order to calculate the distance between them. Second, the formula only works in Euclidean space, which means that all of the points must be laid out on a flat surface. Finally, the formula can be used to find the distance between any two points, as long as you know their coordinates.
To use the Euclidean distance formula, start by identifying the coordinates of both points. Once you have those, plug them into the formula and solve for “d”. The resulting number is the distance between the two points.
Keep in mind that the Euclidean distance formula is just one way to calculate distances between points. There are other formulas that may be more appropriate for certain situations, so it’s important to know when to use each one.
Conclusion
In conclusion, the Euclidean distance formula is a mathematical way of determining the distance between two points in space. It is useful in many fields, such as physics and engineering, because it allows for quick and accurate calculations. The formula can be used in two or three dimensions, and is relatively simple to learn and use. With a little practice, you’ll be able to calculate distances like a pro!
