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Three Dimensional Geometry

Madhya Pradesh Board · Class 12 · Mathematics

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

110 questions40 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

1. Direction Cosines and Direction Ratios

  • Direction cosines are the cosines of the angles a directed line makes with the x, y, and z-axes.
  • If the direction cosines of a line are l, m, n, then l^2 + m^2 + n^2 = 1.
  • Reversing the direction of a line reverses the signs of all its direction cosines.
2

2. Equation of a Line in Space

  • A line in space is uniquely determined if it passes through a given point and has a given direction.
  • A line in space is also uniquely determined if it passes through two given points.
  • If a line passes through point A with position vector a and is parallel to vector b, its vector equation is r = a + λb.
3

3. Angle Between Two Lines

  • The angle between two lines is found using their direction ratios or direction cosines.
  • For two lines with direction ratios a1,b1,c1 and a2,b2,c2, use the dot-product formula for cosine.
  • For direction cosines l1,m1,n1 and l2,m2,n2, the formula becomes very simple.
4

4. Shortest Distance Between Two Lines

  • If two lines intersect, the shortest distance between them is zero.
  • If two lines are parallel, the shortest distance is the perpendicular distance between them.
  • Skew lines are lines that are neither parallel nor intersecting and are non-coplanar.

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Full Notes

Key Concepts

Direction cosines are the cosinesAny three numbers proportional toIfA unique line passes through twoA line is uniquely determined if

Frequently Asked Questions

What are the important topics in Three Dimensional Geometry for Madhya Pradesh Board Class 12 Mathematics?
Three Dimensional Geometry covers several key topics that are frequently asked in Madhya Pradesh 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 — Madhya Pradesh Board Class 12 Mathematics?
Understand the core concepts first, then work through the 110 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

Sources & Official References

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

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