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Three Dimensional Geometry

Karnataka Board · Class 12 · Mathematics

Quick revision notes for Three Dimensional Geometry — Karnataka Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

18 questions20 flashcards5 concepts

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A 3D Cartesian coordinate system showing x, y, and z axes with a directed line L passing through the origin. The angles alpha, beta, and gamma are shown between line L and the positive x, y, and z axe
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Key Topics to Revise

1

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
2

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
3

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°
4

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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Full Notes

Key Concepts

Direction cosines (lVector formFor lines with direction ratios (a₁Lines that are neither parallel norFor skew lines r⃗ = a⃗₁

Frequently Asked Questions

What are the important topics in Three Dimensional Geometry for Karnataka Board Class 12 Mathematics?
Three Dimensional Geometry covers several key topics that are frequently asked in Karnataka Board Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Three Dimensional Geometry — Karnataka Board Class 12 Mathematics?
Start by understanding all key concepts. Practise previous year questions from this chapter. Revise formulas and definitions regularly. Use flashcards for quick revision before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

More resources for Three Dimensional Geometry

All 13 chapters — Karnataka Board Class 12 Mathematics Revision Notes

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