The Distance and Midpoint Formulas in 3D
Simple Explanation
Both the familiar 2D distance and midpoint formulas extend directly to three dimensions, simply by including a third term for the z-coordinates.
Why Do We Need It?
Measuring how far apart two points are, or finding the point exactly between them, is a basic building block used throughout the rest of this chapter and in later applications.
Formula
Distance and Midpoint in Three Dimensions
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²], M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
Both the 2D distance and midpoint formulas extend directly to 3D, simply by including a third (z) term.
- (x₁,y₁,z₁), (x₂,y₂,z₂)
- — the coordinates of the two points
- M
- — the midpoint of the segment joining the two points
When to use it: Whenever the distance between two points in 3D space, or the point exactly halfway between them, is needed.
Worked Example
Find the distance between two points in 3D
Find the distance between A(1,2,3) and B(4,6,15).
Why Does This Work?
The 3D distance formula is really the Pythagorean theorem applied twice — first within the (x,y) plane to find the flat horizontal distance, and then again combining that horizontal distance with the vertical (z) difference into an overall straight-line 3D distance.
Real-Life Example
Finding the straight-line distance between two drones
Two drones are flying at different heights and different horizontal positions.
The 3D distance formula gives their actual straight-line separation directly, accounting for the height difference as well as the horizontal distance.
Practice
Find the midpoint's z-coordinate for A(2,0,4) and B(6,8,10).
MediumCommon mistake
Forgetting the third (z) term entirely, and applying only the plain 2D distance or midpoint formula.
Quick Review
- d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].
- Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).
- Both formulas are the familiar 2D versions, extended with a third coordinate.