Mathematics.

coordinate geometry

Three-Dimensional Coordinates

Analytic Geometry25 minDifficulty3 out of 10

You should know: coordinate plane

Overview

Three-dimensional coordinates extend the familiar (x, y) plane by adding a third mutually perpendicular axis, z, so every point in space is located by an ordered triple (x, y, z). The three coordinate axes divide space into eight octants, analogous to the four quadrants of the plane. Distance and midpoint formulas generalize directly from two dimensions by adding a z-term, since the underlying logic — the Pythagorean theorem applied one axis at a time — is unchanged; this framework underlies 3D graphics, robotics, physics simulations, and the study of surfaces and solids in space.

Intuition

Adding a third axis is like stacking a whole family of 2D planes on top of each other along the z-direction; a point in space is a point in one of those planes, located by its z-value telling you which plane, and its (x,y) telling you where in that plane. The 3D distance formula is proved the same way as the 2D one: build a right triangle from a projected 2D distance and the vertical z-difference, and apply the Pythagorean theorem once more — √((2D distance)² + Δz²) expands exactly to the three-term formula.

Formal Definition

Definition

For points P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) in three-dimensional space:

d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}
3D distance formula
M=(x1+x22, y1+y22, z1+z22)M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2},\ \frac{z_1+z_2}{2}\right)
3D midpoint formula
(x−h)2+(y−k)2+(z−l)2=r2(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2
Sphere of radius r centered at (h,k,l)

Worked Examples

  1. Apply the 3D distance formula.

    d=(4−1)2+(6−2)2+(3−3)2=9+16+0=25=5d = \sqrt{(4-1)^2+(6-2)^2+(3-3)^2} = \sqrt{9+16+0} = \sqrt{25} = 5

Answer: d = 5 (the z-coordinates are equal, so this reduces to a 2D distance in the plane z=3)

Practice Problems

Difficulty 3/10

Find the distance between (0, 0, 0) and (3, 4, 12).

Difficulty 4/10

Find the midpoint of the segment from (−2, 5, 1) to (6, −1, 7).

Difficulty 6/10

A drone flies from position (0, 0, 10) to (30, 40, 10) meters (constant altitude). Find the horizontal distance traveled and confirm the altitude term contributes nothing to that distance.

Quiz

The 3D distance formula between (x₁,y₁,z₁) and (x₂,y₂,z₂) is:
Three-dimensional space is divided by the three coordinate planes into how many octants?
The equation (x−h)²+(y−k)²+(z−l)²=r² describes:

Summary

  • 3D coordinates (x,y,z) locate points in space using three mutually perpendicular axes, dividing space into 8 octants.
  • Distance and midpoint formulas extend directly from 2D by adding a z-coordinate term, both still rooted in the Pythagorean theorem.
  • The sphere equation (x−h)²+(y−k)²+(z−l)²=r² is the natural 3D generalization of the circle equation.

References

  1. WebsiteWikipedia — Cartesian coordinate system