Three-Dimensional Geometry: Distance in 3D Space
Explore three-dimensional geometry concepts through visualizing points in 3D space. Calculate the distance between two points using the 3D distance formula and understand how coordinates relate to position in three-dimensional space.
Help & Instructions
▼- View 3D Space: See the 3D coordinate system with origin at the corner
- Place Points: Click on the 3D visualization to place points A and B
- Calculate Distance: Click to calculate the distance between the two points
- Adjust Points: Move points to see how distance changes
- Analyze Formula: View the step-by-step calculation using the 3D distance formula
- Understand the 3D coordinate system
- Learn to calculate distance between two points in 3D space
- Visualize how coordinates determine position in 3D
- Recognize real-world applications of 3D geometry
3D Space Visualization
Visualization of 3D coordinate system with points:
Distance Calculation
Distance between Point A and Point B:
Point Coordinates
Coordinates of points in 3D space:
3D Distance Formula
Step-by-step calculation:
The distance between two points in three-dimensional space is calculated using an extension of the Pythagorean theorem. In 3D space, we need to account for differences in all three coordinates (x, y, and z) to find the straight-line distance between two points.
The Mathematics of 3D Distance
The distance between two points in three-dimensional space is the length of the straight line connecting them. It is calculated using the 3D distance formula, which is an extension of the Pythagorean theorem to three dimensions.
Where (x₁, y₁, z₁) and (x₂, y₂, z₂) are the coordinates of the two points in 3D space.
- Calculate the difference in x-coordinates: Δx = x₂ - x₁
- Calculate the difference in y-coordinates: Δy = y₂ - y₁
- Calculate the difference in z-coordinates: Δz = z₂ - z₁
- Square each difference: (Δx)², (Δy)², (Δz)²
- Sum the squares: (Δx)² + (Δy)² + (Δz)²
- Take the square root of the sum: √[(Δx)² + (Δy)² + (Δz)²]
3D distance calculations are used in:
- Architecture: Measuring distances in 3D building models
- Engineering: Calculating distances between components in 3D space
- Computer Graphics: Rendering 3D scenes and calculating object distances
- Robotics: Path planning and obstacle avoidance in 3D environments
- Physics: Calculating distances between particles in 3D space
