Andromeda
Note

Distance Between Points in Space

Definition

The formula for the distance between two points in space is an extension of the Pythagorean theorem into three dimensions.

Why It Matters

Depth is the variable that transforms a flat map into a real, navigable world. This formula is the essential “tape measure” for everything from calculating the clearance of engine parts to the trajectory of a spacecraft, providing the mathematical anchor for building and moving in 3D space.

Core Concepts

  • Distance Formula The distance between two points P1(x1,y1,z1)P_1(x_1, y_1, z_1) and P2(x2,y2,z2)P_2(x_2, y_2, z_2) is given by:

∣P1P2∣=(x2−x1)2+(y2−y1)2+(z2−z1)2|P_1P_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

  • How to read: “The length of the segment P one P two equals the square root of the quantity x two minus x one squared, plus the quantity y two minus y one squared, plus the quantity z two minus z one squared.”

  • Meaning: 3D Pythagorean theorem—straight-line Euclidean distance from the displacement in each axis.

  • Context This formula represents the length of the line segment joining P1P_1 and P2P_2 and is used to define spheres and other geometric surfaces in R3\mathbb{R}^3.

Connected Concepts

Connected notes