Learning Objectives
- Understand direction cosines and direction ratios
- Learn equations of lines and planes in 3D
- Study angle between lines, angle between planes
- Calculate distance between lines and from point to plane
Key Concepts
Direction Cosines and Ratios
Direction cosines (l, m, n): Cosines of angles made with x, y, z axes. l2 + m2 + n2 = 1. Direction ratios (a, b, c): Any numbers proportional to direction cosines. l = a/√(a2+b2+c2), etc. Direction ratios of line joining (x₁,y₁,z₁) and (x₂,y₂,z₂): (x₂-x₁, y₂-y₁, z₂-z₁).
Angle between two lines with direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂): cosθ = |a₁a₂+b₁b₂+c₁c₂| / (√(a₁2+b₁2+c₁2) × √(a₂2+b₂2+c₂2)). Perpendicular if a₁a₂+b₁b₂+c₁c₂ = 0. Parallel if a₁/a₂ = b₁/b₂ = c₁/c₂.
Equation of a Line
Vector form: r = a + λb (passes through point with position vector a and parallel to b). Cartesian form: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c (passing through (x₁,y₁,z₁) with direction ratios a, b, c). Two-point form: (x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁).
Equation of a Plane
Vector form: r · n = d (normal form). Cartesian form: ax + by + cz + d = 0 where (a, b, c) are direction ratios of normal. Intercept form: x/a + y/b + z/c = 1. Through three non-collinear points: Use determinant form.
Important Distances
Distance of point (x₁,y₁,z₁) from plane ax+by+cz+d = 0: D = |ax₁+by₁+cz₁+d| / √(a2+b2+c2).
Angle between two planes: cosθ = |a₁a₂+b₁b₂+c₁c₂| / (√(a₁2+b₁2+c₁2) × √(a₂2+b₂2+c₂2)).
Angle between a line and a plane: sinα = |b·n| / (|b||n|) where b is direction of line, n is normal to plane.
Shortest distance between skew lines: r = a₁ + λb₁ and r = a₂ + μb₂: d = |(a₂ - a₁) · (b₁ × b₂)| / |b₁ × b₂|. If b₁ × b₂ = 0, lines are parallel; distance = |(a₂-a₁) × b| / |b|.
Coplanar lines: (a₂ - a₁) · (b₁ × b₂) = 0 (shortest distance = 0).
Summary
3D geometry uses direction cosines/ratios to specify orientations. Lines are expressed in vector or Cartesian form. Planes are defined by their normal vectors. Key calculations include distances between points and planes, angles between geometric objects, and shortest distance between skew lines.
Important Terms
- Skew lines: Non-parallel, non-intersecting lines in 3D
- Direction cosines: l, m, n with l2+m2+n2 = 1
- Normal to plane: Vector perpendicular to the plane
- Foot of perpendicular: Point on plane closest to a given point
Quick Revision
- Line: r = a + λb; Cartesian: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
- Plane: r·n = d; Cartesian: ax + by + cz + d = 0
- Distance from point to plane: |ax₁+by₁+cz₁+d|/√(a2+b2+c2)
- Skew line distance: |(a₂-a₁)·(b₁×b₂)|/|b₁×b₂|
- Line ⊥ plane: direction ratios proportional to normal
- Line ∥ plane: b·n = 0
- Coplanar lines: scalar triple product = 0