3-D Point Distance from Origin
Three-Dimensional Geometry · Class XII
Slide x, y, z components and read off the Euclidean distance √(x²+y²+z²).
- Distance from O3
- Direction cosine l0.3333
- Direction cosine m0.6667
- Direction cosine n0.6667
- Distance as z slides
Formulas in this lab
- Distance from O
- Direction cosine l
- Direction cosine m
- Direction cosine n
Frequently asked questions
▶What is the distance from the origin to $(x, y, z)$ in 3-D?
By Pythagoras in three dimensions, $d = \sqrt{x^2 + y^2 + z^2}$. So $(1, 2, 2)$ is $\sqrt{1+4+4} = 3$ units from the origin. This generalizes the 2-D distance formula by adding the $z^2$ term.
▶How do I use the 3-D distance lab?
Slide the coordinates $(x, y, z)$. The lab returns $|\vec r|$ and the direction cosines $(x/|\vec r|, y/|\vec r|, z/|\vec r|)$. Try $(2, 3, 6)$: distance is $7$, a clean integer triple in 3-D.
▶How does this preview the formula for distance between two 3-D points?
Replace $x, y, z$ by differences $(x_2 - x_1), (y_2 - y_1), (z_2 - z_1)$: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. The origin case is just $(x_1, y_1, z_1) = (0, 0, 0)$. Master the origin case first; the general one slots in.
▶Where does 3-D distance show up in real applications?
GPS uses distances from your phone to four satellites to triangulate position. Game engines render 3-D scenes by computing camera-to-object distances. Even drone delivery routing computes straight-line distances in 3-D first, then adjusts for obstacles.