Learning Objectives
- Understand coordinate axes and planes in 3D space
- Learn distance formula and section formula in 3D
- Study octants and coordinates in three dimensions
Key Concepts
Coordinate System in 3D
Three mutually perpendicular axes: x, y, z (right-hand system). A point P has coordinates (x, y, z). Coordinate planes: xy-plane (z = 0), yz-plane (x = 0), xz-plane (y = 0). Eight octants: determined by signs of x, y, z. Origin O = (0, 0, 0).
Distance from coordinate planes: Distance of (x, y, z) from xy-plane = |z|; from yz-plane = |x|; from xz-plane = |y|. Distance from axes: Distance from x-axis = √(y2 + z2); from y-axis = √(x2 + z2); from z-axis = √(x2 + y2).
Distance Formula
Distance between P(x₁,y₁,z₁) and Q(x₂,y₂,z₂): PQ = √[(x₂-x₁)2 + (y₂-y₁)2 + (z₂-z₁)2]. Distance from origin: √(x2 + y2 + z2).
Section Formula
Point dividing line segment joining P(x₁,y₁,z₁) and Q(x₂,y₂,z₂) in ratio m:n internally: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)). Externally: replace n by -n. Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).
Centroid of triangle with vertices (x₁,y₁,z₁), (x₂,y₂,z₂), (x₃,y₃,z₃): ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3).
Octant Signs
I octant: (+,+,+), II: (-,+,+), III: (-,-,+), IV: (+,-,+), V: (+,+,-), VI: (-,+,-), VII: (-,-,-), VIII: (+,-,-).
Summary
Three-dimensional geometry extends 2D concepts to three mutually perpendicular axes. The distance formula, section formula, and midpoint formula are natural extensions of their 2D counterparts. Space is divided into eight octants by the three coordinate planes.
Important Terms
- Octant: One of eight regions created by coordinate planes in 3D
- Coordinate planes: xy-plane, yz-plane, xz-plane dividing space
- Section formula: Divides line segment in given ratio in 3D
- Centroid: Point of intersection of medians; averages of coordinates
Quick Revision
- Distance = √[(x₂-x₁)2 + (y₂-y₁)2 + (z₂-z₁)2]
- Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
- Distance from xy-plane = |z|; from yz-plane = |x|; from xz-plane = |y|
- Distance from x-axis = √(y2+z2)
- Centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)
- Eight octants determined by signs of (x, y, z)