Class 12 Mathematics — Chapter 5: THREE DIMENSIONAL GEOMETRY
Chapter 5: THREE DIMENSIONAL GEOMETRY is a chapter in Class 12 Mathematics (Part 2), part of the CBSE NCERT curriculum followed by over 25 million students across India. This chapter covers 7 topics including Direction Cosines (d.c.'s), Direction Ratios (d.r.'s), Calculating Direction Cosines from Direction Ratios. BrainWeave provides free AI-powered explanations — by voice or text, in Hindi or English — with no signup required.
What you'll learn
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▸Direction Cosines (d.c.'s)Core conceptdirection cosinesdirection anglescos αl, m, nl²+m²+n²=1
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▸Direction Ratios (d.r.'s)Core conceptdirection ratiosproportionaldirection numbersa, b, cλl, λm, λn
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▸Calculating Direction Cosines from Direction RatiosCore conceptconvertformulanormalize√(a²+b²+c²)plus-minus
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▸Direction Ratios & Cosines from Two PointsCore concepttwo pointsline segmentx₂-x₁PQdistance formula
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▸Collinearity of Points using Direction Ratioscollinearproportionalparallel vectorscommon pointshow that
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▸Vector Equation of a LineCore conceptvector equationr = a + λbposition vectorparallel vectorparameter λ
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▸Cartesian Equation of a LineCore conceptcartesian equationsymmetric form(x-x₁)/aparametric equationseliminating parameter
Chapter Summary
Understanding that direction cosines (l, m, n) of a directed line are the cosines of the angles (α, β, γ) it makes with the positive x, y, and z axes, respectively. A key property is that the sum of their squares is always one (l² + m² + n² = 1).
Defining direction ratios (a, b, c) as any three numbers that are proportional to the direction cosines of a line. Unlike direction cosines, a line can have infinitely many sets of direction ratios.
Learning the method to find the direction cosines of a line when its direction ratios (a, b, c) are given, using the formulas l = ±a/√(a²+b²+c²), m = ±b/√(a²+b²+c²), and n = ±c/√(a²+b²+c²).
Determining the direction ratios (x₂-x₁, y₂-y₁, z₂-z₁) and subsequently the direction cosines for a line segment that passes through two given points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).
Applying the concept of direction ratios to verify if three given points (A, B, C) are collinear. This is done by showing that the direction ratios of the line segments AB and BC are proportional.
Understanding and formulating the vector equation of a line, r = a + λb, given that it passes through a point with position vector 'a' and is parallel to a given vector 'b'.
Deriving the Cartesian or symmetric form of the equation of a line, (x-x₁)/a = (y-y₁)/b = (z-z₁)/c, from its vector form, where (x₁, y₁, z₁) is a point on the line and (a, b, c) are its direction ratios.
Practice Questions from this Chapter
Tap "Get Solution" on any question to ask our AI tutor.
- Define direction cosines simply. Get Solution →
- Visualize a line in space. Get Solution →
- Calculate a 3D distance. Get Solution →
- What are the cosines of the direction angles α, β, and γ of a directed line called? Get Solution →
- If l, m, and n are the direction cosines of a line, what is the value of l² + m² + n²? Get Solution →
- Any three numbers that are proportional to the direction cosines of a line are called what? Get Solution →
- What are the direction cosines of the x-axis? Get Solution →
- What is the vector equation of a line passing through a point with position vector `a` and parallel to vector `b`? Get Solution →
Did you know?
- 💡 Our universe might actually possess more than three spatial dimensions we perceive.
- 💡 A tesseract, a four-dimensional cube, is truly impossible for humans to visualize.
- 💡 Ancient Egyptians utilized advanced three-dimensional geometry to construct their colossal pyramids.
- 💡 Bees navigate using polarized light and the sun, creating complex three-dimensional maps.
- 💡 The shortest path between two points in curved space is not always a straight line.
Frequently Asked Questions
How many topics are covered in this chapter?
This chapter covers 7 key topics: Direction Cosines (d.c.'s), Direction Ratios (d.r.'s), Calculating Direction Cosines from Direction Ratios, Direction Ratios & Cosines from Two Points, Collinearity of Points using Direction Ratios, and more. The BrainWeave AI tutor explains each one with examples.
Is Chapter 5: THREE DIMENSIONAL GEOMETRY important for board exams?
Yes — Class 12 is a CBSE board exam year, and every NCERT chapter is part of the syllabus. Use BrainWeave's AI tutor to master this chapter, then practice with the auto-generated quizzes and mind maps.
Can I get NCERT solutions for this chapter in Hindi?
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