Distance, midpoint and centroid

The formulae of the plane extend simply by adding the third coordinate.

Property — Distance, midpoint, centroid

Given $P(x_P,y_P,z_P)$ and $Q(x_Q,y_Q,z_Q)$:

• Distance: $d(P,Q) = \sqrt{(x_Q-x_P)^2 + (y_Q-y_P)^2 + (z_Q-z_P)^2}$  (Pythagoras applied twice).
• Midpoint: $M = \left(\dfrac{x_P+x_Q}{2},\;\dfrac{y_P+y_Q}{2},\;\dfrac{z_P+z_Q}{2}\right)$.
• Centroid of a triangle $ABC$: $G = \left(\dfrac{x_A+x_B+x_C}{3},\;\dfrac{y_A+y_B+y_C}{3},\;\dfrac{z_A+z_B+z_C}{3}\right)$.

The distance formula is the real novelty: applying Pythagoras’ theorem first in the horizontal plane and then vertically gives the square root of the sum of the squares of the three coordinate differences.

Links

Topics: Analytic geometry in space
Concepts: Centroid · Cartesian coordinates · Distance between points · Cartesian plane · Midpoint
Skills: Analytic geometry · Using formulae