Three Dimensional Geometry
Rajasthan Board · Class 12 · Mathematics
Quick revision notes for Three Dimensional Geometry — Rajasthan Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Direction Cosines and Direction Ratios
- Direction cosines are cosines of angles made by a line with positive x, y, and z axes
- Direction cosines are denoted by l, m, n and satisfy l² + m² + n² = 1
- A line can be extended in two directions, giving two sets of direction cosines
Equation of a Line in Space
- A line is uniquely determined by a point and direction, or by two points
- Vector form: r⃗ = a⃗ + λb⃗ where a⃗ is position vector of point, b⃗ is direction vector
- Cartesian form: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
Angle Between Two Lines
- Angle between lines is angle between their direction vectors
- If lines don't pass through origin, draw parallel lines through origin
- Acute angle θ satisfies 0° ≤ θ ≤ 90°
Shortest Distance Between Lines
- Distance is zero if lines intersect
- For parallel lines, distance is perpendicular distance between them
- For skew lines, shortest distance is along common perpendicular
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Three Dimensional Geometry covers several key topics that are frequently asked in Rajasthan Board Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
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