Vector Distance Calculator

Calculator and formulas for computing Euclidean distance between two vectors

Vector Distance Calculator

Euclidean Distance Between Two Points

Calculates the Euclidean distance between two vectors through difference calculation: d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

Select Vector Dimension
Plane
Space
Four coordinates
First Point (P₁)
Second Point (P₂)
Distance Result
Distance d:
Calculation: d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

Vector Distance Info

Distance Properties

Symmetric: d(A,B) = d(B,A)

d ≥ 0 Triangle Inequality Euclidean

Geometric: Shortest path between points
Formula: Difference squares summed and rooted

Examples
d([0,0], [3,4]) = √(3²+4²) = 5
d([1,2,3], [4,6,8]) = √(3²+4²+5²) = √50
d([0,0,0], [0,0,0]) = 0 (same points)

Formulas for Vector Distance

General Formula
\[d = \sqrt{\sum_{i=1}^n (p_{2i} - p_{1i})^2}\]

n-dimensional Euclidean metric

2D Distance
\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]

Euclidean distance in the plane

3D Distance
\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]

Spatial Euclidean distance

4D Distance
\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 + (w_2-w_1)^2}\]

Higher-dimensional distance

Calculation Examples for Vector Distance

Example 1: 2D Distance
P₁ = (0, 0), P₂ = (3, 4)
\[d = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]

Classic 3-4-5 triangle

Example 2: 3D Distance
P₁ = (0, -2, 7), P₂ = (8, 4, 3)
\[\begin{aligned} d &= \sqrt{(8-0)^2 + (4-(-2))^2 + (3-7)^2} \\ &= \sqrt{64 + 36 + 16} = \sqrt{116} ≈ 10.77 \end{aligned}\]

Spatial distance calculation

Step-by-Step Calculation
Form Differences
Δx, Δy, Δz
Square Values
(Δx)², (Δy)², (Δz)²
Sum & Root
√(Σ Δᵢ²)

Systematic approach for all dimensions

Applications of Vector Distance

Euclidean distance is fundamental for many areas of science and technology:

Navigation & GPS
  • Distance calculations between coordinates
  • Route optimization and path planning
  • Geocaching and location services
  • Great circle distances
Machine Learning
  • K-Nearest-Neighbor algorithm
  • Clustering (K-Means, hierarchical)
  • Similarity measurements
  • Feature space distances
Computer Graphics
  • 3D object distances and collision detection
  • Ray-tracing and lighting
  • Level-of-detail calculations
  • Camera and view distances
Science
  • Physics: Particle distances and force calculations
  • Astronomy: Distances in space
  • Chemistry: Molecular distances
  • Biology: Genetic distances

Vector Distance: The Euclidean Metric

The Euclidean distance is the natural generalization of the distance between two points to arbitrary dimensions. This fundamental metric forms the foundation for countless applications - from GPS navigation through machine learning to scientific research. It satisfies all properties of a mathematical metric and corresponds to our intuitive understanding of "distance".

Summary

Vector distance combines geometric intuition with practical applicability. The Pythagorean formula - square root of the difference squares - provides exact distance measurements in arbitrary dimensions. From 2D navigation through 3D graphics to high-dimensional data analysis, Euclidean distance remains the universal measure for similarity and proximity. It demonstrates how mathematical elegance solves practical problems in science and technology.




Vector Functions

Addition • Subtraction • Multiplication • Scalar Multiplication • Division • Scalar Division • Dot Product • Cross Product • Interpolation • Distance • Distance Squaret • Normalization • Reflection • Magnitude • Squared-Magnitude • Triple-Product