Three-Dimensional Distance Calculator
Calculate the distance between two points in 3D space using coordinates (x, y, z)
Calculate 3D Distance
First Point (x₁, y₁, z₁)
Second Point (x₂, y₂, z₂)
3D Distance Result
Step-by-Step Calculation
Formula: D = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
Step 1: Calculate differences:
x₂ - x₁ = 0 - 0 = 0.000
y₂ - y₁ = 0 - 0 = 0.000
z₂ - z₁ = 0 - 0 = 0.000
Step 2: Square the differences:
(0.000)² = 0.000000
(0.000)² = 0.000000
(0.000)² = 0.000000
Step 3: Sum the squares:
0.000000 + 0.000000 + 0.000000 = 0.000000
Step 4: Take the square root:
D = √0.000000 = 0.000000
Example Calculations
Example 1: Distance from Origin
Point 1: (0, 0, 0) - Origin
Point 2: (1, 1, 1)
Calculation: D = √[(1-0)² + (1-0)² + (1-0)²]
Result: D = √[1 + 1 + 1] = √3 ≈ 1.732
Example 2: Complex Points
Point 1: (1, 1, 1)
Point 2: (3, 6, 9)
Calculation: D = √[(3-1)² + (6-1)² + (9-1)²]
Result: D = √[4 + 25 + 64] = √93 ≈ 9.643
3D Distance Formula
D: Distance between points
(x₁,y₁,z₁): First point coordinates
(x₂,y₂,z₂): Second point coordinates
Calculation Steps
Find coordinate differences
x₂-x₁, y₂-y₁, z₂-z₁
Square each difference
Square all three values
Sum the squares
Add all squared differences
Take square root
Final distance result
Quick Tips
Distance is always positive (scalar value)
Origin is at coordinates (0, 0, 0)
Order of points doesn't affect distance
Distance from point to itself is 0
Understanding 3D Distance Calculation
What is 3D Distance?
The three-dimensional distance is the straight-line distance between two points in 3D space. It's the shortest path connecting two points, regardless of any obstacles that might be in between.
Key Properties:
- •Always positive: Distance is a scalar quantity
- •Symmetric: Distance from A to B equals distance from B to A
- •Triangle inequality: Direct path is always shortest
Mathematical Foundation
The 3D distance formula is an extension of the Pythagorean theorem to three dimensions. It's derived from the concept of Euclidean distance in multi-dimensional space.
Formula Derivation:
1. Start with two points: P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂)
2. Create a displacement vector: (x₂-x₁, y₂-y₁, z₂-z₁)
3. Calculate vector magnitude: √[component₁² + component₂² + component₃²]
4. This gives us the 3D distance formula
Applications and Uses
Engineering & Physics
Calculate distances between objects in 3D space, trajectory analysis, and spatial measurements in mechanical design.
Computer Graphics
Determine distances between 3D models, collision detection, and spatial relationships in virtual environments.
Navigation & GPS
Calculate straight-line distances considering altitude differences, aviation navigation, and satellite positioning.