Three Dimensional Geometry Class 12 Maths Notes – Formulas & Quick Revision

3D Geometry mein direction cosines, lines ke equations aur unke beech angle/distance aate hain. Yahan aapko is chapter ke sabse important formulas aur quick revision points free milte hain — NCERT/CBSE syllabus ke hisaab se, boards aur JEE/NEET dono ke liye.

Important Formulas & Concepts — Three Dimensional Geometry

• Direction cosines: l² + m² + n² = 1
• l = cosα, m = cosβ, n = cosγ
• Line: r = a + λb (vector form)
• Cartesian: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
• Angle: cosθ = |b₁·b₂|/(|b₁||b₂|)
• Shortest distance (skew lines): |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|
• Parallel lines distance: |(a₂−a₁)×b|/|b|
• Perpendicular lines: b₁·b₂ = 0

Quick Revision Points

• Skew lines ka shortest distance 5-marker classic hai
• DC aur DR ka fark samjho (ratios vs cosines)
• Line ke vector aur cartesian forms mein convert karna aana chahiye
• Rationalised syllabus mein plane hat gaya hai — lines par focus
• Point se line ki distance foot of perpendicular se

FAQ

Shortest distance ka formula bada lagta hai?

PDF mein step-by-step breakdown hai — b₁×b₂ pehle, phir dot, phir modulus.

Plane ab syllabus mein hai?

Rationalised CBSE syllabus se plane hata hai — par JEE ke liye padhna hoga.

Explore more: free sample notes, free calculators aur interactive simulators.