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Euclidean Distance

Last Updated : 21 May, 2024
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Euclidean Distance is defined as the distance between two points in Euclidean space. To find the distance between two points, the length of the line segment that connects the two points should be measured.

In this article, we will explore what is Euclidean distance, the Euclidean distance formula, its derivation, examples, etc.

Euclidean-Distance

What is Euclidean Distance?

The measure which gives the distance between any two points in an n-dimensional plane is known as Euclidean Distance. Euclidean distance between two points in the Euclidean space is defined as the length of the line segment between two points.

Euclidean distance is like measuring the straightest and shortest path between two points. Imagine you have a string and you stretch it tight between two points on a map; the length of that string is the Euclidean distance. It tells you how far apart the two points are without any turns or bends, just like a bird would fly directly from one spot to another.

Euclidean Distance Formula

Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space; the Euclidean Distance between them is given by using the formula:

d = √[(x2 – x1)2 + (y2 – y1)2]

Where,

  • d is Euclidean Distance
  • (x1, y1) is Coordinate of the first point
  • (x2, y2) is Coordinate of the second point

Euclidean Distance in 3D

If the two points (x1, y1, z1) and (x2, y2, z2) are in a 3-dimensional space, the Euclidean Distance between them is given by using the formula:

d = √[(x2 – x1)2 + (y2 – y1)2+ (z2 – z1)2]

where,

  • d is Euclidean Distance
  • (x1, y1, z1) is Coordinate of the first point
  • (x2, y2, z2) is Coordinate of the second point

Euclidean Distance in nD

In general, the Euclidean Distance formula between two points (x11, x12, x13, …., x1n) and (x21, x22, x23, …., x2n) in an n-dimensional space is given by the formula:

d = √[∑(x2i – x1i)2]

Where,

  • i Ranges from 1 to n
  • d is Euclidean distance
  • (x11, x12, x13, …., x1n) is Coordinate of First Point
  • (x21, x22, x23, …., x2n) is Coordinate of Second Point

Euclidean Distance Formula Derivation

Euclidean Distance Formula is derived by following the steps added below:

Step 1: Let us consider two points, A (x1, y1) and B (x2, y2), and d is the distance between the two points.

Step 2: Join the points using a straight line (AB).

Step 3: Now, let us construct a right-angled triangle whose hypotenuse is AB, as shown in the figure below.

Distance Between Two Points

Step4: Now, using Pythagoras theorem we know that,

(Hypotenuse)2 = (Base)2 + (Perpendicular)2

d2 = (x2 – x1)2 + (y2 – y1)2

Now, take the square root on both sides of the equation, we get

d = √(x2 – x1)2 + (y2 – y1)2

Euclidean Distance and Manhattan Disctance

Differences between the Euclidean and Manhattan methods of measuring distance are listed in the following table:

Aspect Euclidean Distance Manhattan Distance
Definition Measures the shortest straight-line distance between two points. Measures the distance between two points along axes at right angles.
Formula (2D) d = √[(x2 – x1)2 + (y2 – y1)2] d = [|x2 – x1| + |y2 – y1|]
Path Direct straight line. Path that resembles city blocks or a grid pattern.
Metric Name L2 norm or Euclidean norm. L1 norm or Manhattan norm.
Use Cases Used in scenarios where direct distances are needed (e.g., physics). Commonly used in planning, urban design, and certain optimization algorithms.
Sensitivity to Scaling Less sensitive to scaling of axes. More sensitive to scaling of axes since it adds absolute differences.

Conclusion

Euclidean Distance is a metric for measuring the distance between two points in Euclidean space, reflecting the length of the shortest path connecting them, which is a straight line. The formula for calculating Euclidean Distance depends on the dimensionality of the space. In a 2-dimensional plane, the distance d between points is, d = d = √[(x2 – x1)2 + (y2 – y1)2]. In 3D, d = √[(x2 – x1)2 + (y2 – y1)2+ (z2 – z1)2].

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Solved Questions on Euclidean Distance

Here are some sample problems based on the distance formula.

Question 1: Calculate the distance between the points (4,1) and (3,0).

Solution:

Using Euclidean Distance Formula:

d = √(x2 – x1)2 + (y2 – y1)2

d = √(3 – 4)2 + (0 – 1)2

⇒ d = √(1 + 1)

⇒ d = √2 = 1.414 unit

Question 2: Show that the points A (0, 0), B (4, 0), and C (2, 2√3) are the vertices of an Equilateral Triangle.

Solution:

To prove that these three points form an equilateral triangle, we need to show that the distances between all pairs of points, i.e., AB, BC, and CA, are equal.

Distance between points A and B:

AB = √[(4– 0)2 + (0-0)2]

⇒ AB = √16

AB = 4 unit

Distance between points B and C:

BC = √[(2-4)^2 + (2√3-0)^2]

⇒ BC = √[4+12] = √16

BC = 4 unit

Distance between points C and A:

CA = √[(0-2)2 + (0-2√3)2]

⇒ CA = √[4 + 12] = √16

CA = 4 unit

Here, we can observe that all three distances, AB, BC, and CA, are equal.

Therefore, the given triangle is an Equilateral Triangle

Question 3: Mathematically prove Euclidean distance is a non negative value.

Solution:

Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space; the Euclidean Distance between them is given by using the formula:

d = √(x2 – x1)2 + (y2 – y1)2

We know that squares of real numbers are always non-negative.

⇒(x2 – x1)2 >= 0 and (y2 – y1)2 >= 0

√(x2 – x1)2 + (y2 – y1)2 >= 0

As square root of a non-negative number gives a non-negative number,

Therefore Euclidean distance is a non-negative value. It cannot be a negative number.

Question 4: A triangle has vertices at points A(2, 3), B(5, 7), and C(8, 1). Find the length of the longest side of the triangle.

Solution:

Given, the points A(2, 3), B(5, 7), and C(8, 1) are the vertices of a triangle.

Distance between points A and B:

AB = √[(5-2)2 + (7-3)2]

⇒ AB = √9+16= √25

AB = 5 unit

Distance between points B and C:

BC = √[(8-5)2 + (1-7)2]

⇒ BC = √[9+36] = √45

BC = 6.708 unit

Distance between points C and A:

CA = √[(8-2)2 + (1-3)2]

⇒ CA = √[36+4] = √40

CA = 6.325 unit

Therefore, the length of the longest side of triangle is 6.708 unit.

Practice Problems on Euclidean Distance

P1: Calculate the Euclidean distance between points P(1, 8, 3) and Q(6, 6, 8).

P2: A car travels from point A(0, 0) to point B(5, 12). Calculate the distance traveled by the car?

P3: An airplane flies from point P(0, 0, 0) to point Q(100, 200, 300). Calculate the distance traveled by the airplane.

P4: A triangle has vertices at points M(1, 2), N(4, 6), and O(7, 3). Find the perimeter of the triangle.

P5: On a graph with points K(2, 3) and L(5, 7), plot these points and calculate the Euclidean distance between them.

P6: A drone needs to fly from point A(1, 1) to point B(10, 10). Find the shortest path the drone should take to conserve battery?

P7: A robotic arm moves from position J(1, 2, 3) to position K(4, 5, 6). Calculate the total distance traveled by the robotic arm.

FAQs on Euclidean Distance

Define Euclidean Distance.

Euclidean distance measures the straight-line distance between two points in Euclidean space.

What is the distance formula for a 2D Euclidean Space?

Euclidean Distance between two points (x1, y1) and (x2, y2) in using the formula:

d = √[(x2 – x1)2 + (y2 – y1)2]

What are some properties of Euclidean Distance?

  • Euclidean distance is always non-negative because it represents a physical distance in space, which cannot be a negative value.
  • Distance between a point and itself is always zero

Can Euclidean Distance be negative?

Euclidean Distance can’t be negative as it represents a physical distance. It can be a zero value but can’t be a negative value.

How can Euclidean Distance be extended to higher dimensions?

In general, the Euclidean Distance formula between two points (x11, x12, x13, …., x1n) and (x21, x22, x23, …., x2n) in an n-dimensional space is given by the formula:

d = √[∑(x2i – x1i)2]

What is the difference between Euclidean Distance and Manhattan Distance?

Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space;

Euclidean Distance between them is given by using the formula:

d = √[(x2 – x1)2 + (y2 – y1)2], (Calculates the square root of the sum of squared differences)

Manhattan Distance between them is given by using the formula:

d = [|x2 – x1| + |y2 – y1|], (Calculates the distance between two points as the sum of the absolute differences in their coordinates)



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