← GSEB Class 12
Three Dimensional Geometry
Chapter Overview
Direction cosines (l, m, n) and direction ratios (a, b, c) describe line direction. Line equation: vector form r = a + lambda*b, Cartesian form (x-x1)/a = (y-y1)/b = (z-z1)/c. Angle between lines, shortest distance between skew lines. Plane equations: general ax + by + cz + d = 0, intercept, normal, and through three points. Distance from point to plane.
Topics Covered
- Direction Cosines and Ratios
- Line in 3D Space
- Angle Between Lines
- Shortest Distance Between Lines
- Plane Equation
- Plane Forms
- Angle Between Planes
- Distance from Point to Plane
- Angle Between Line and Plane
Key Formulas
cos A = |a1a2 + b1b2 + c1c2|/√sum a2√sum b2
SD = |(a2-a1).(b1xb2)|/|b1xb2|
Dist point-plane = |Ax1 + By1 + Cz1 + D|/√A2 + B2 + C2
Real-World Applications
Applications: 3D modeling, CAD software, architecture, robotics path planning, game development.
Study Tips
Tip: Vector form is often simpler than Cartesian
Tip: Practice cross product for plane normals
Tip: Sketch 3D situations mentally