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Three Dimensional Geometry — Revision Notes

Madhya Pradesh Board · Class 12 · Mathematics

Three Dimensional Geometry revision notes for Madhya Pradesh Board Class 12 Mathematics: 4 topics in quick points.

110 questions40 flashcards18 formulas & key relations5 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

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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.
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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.
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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.
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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?
Key topics in Three Dimensional Geometry include Direction Cosines and Direction Ratios, Equation of a Line in Space, Angle Between Two Lines, Shortest Distance Between Two Lines. Study these first, then practise questions on each for the Madhya Pradesh Board Class 12 board exam.
How should I revise Three Dimensional Geometry for the Madhya Pradesh Board Class 12 board exam?
Learn the core ideas first, then work through the 110 practice questions on Three Dimensional Geometry. Revise 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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All Madhya Pradesh Board Class 12 Mathematics Revision Notes

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