🪐🎲 Chapter 11 · Class 12 Maths

Three-Dimensional Geometry

Three-Dimensional Geometry extends the concepts of coordinate geometry to three dimensions, introducing the idea of points, lines, and planes in space.

Topics

Exercises

Board Marks

Topics Covered

5 Topics

📖

Direction Cosines & Direction Ratios

Direction cosines · Direction ratios · Fundamental Relation

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📊

Equation of a Line in 3D Space

Cartesian form · Vector form · Symmetric form

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🔢

Angle Between Two Lines & shortest Distance

Angle · Distance · Skew Lines

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➕

Equation of a Plane in 3D Space

Cartesian form · Vector form · Normal form

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📝

Advanced Topics in 3D Geometry

Coplanarity · Angle between planes

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Key Formulas

Quick Reference

Direction Cosines

l = cos α, m = cos β, n = cos γ
l² + m² + n² = 1

DR → DC

l = ±a/√(a² + b² + c²)
m = ±b/√(a² + b² + c²)
n = ±c/√(a² + b² + c²)

Vector

r = a + λb
r = a + λ(b - a)

Cartesian Form

(x - x₁)/a = (y - y₁)/b = (z - z₁)/c
(x - x₁)/(x₂ - x₁) = (y - y₁)/(y₂ - y₁) = (z - z₁)/(z₂ - z₁)

Topics Covered

Key Formulas

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