Class 12 Mathematics — Chapter 5: THREE DIMENSIONAL GEOMETRY

What you'll learn

• ▸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
• ▸Vector Equation of a Line
• ▸Cartesian Equation of a Line

Chapter Summary

--- PAGE 1 --- 12080CH11 Chapter 11 THREE DIMENSIONAL GEOMETRY ❖ The moving power of mathematical invention is not reasoning but imagination. – A. DEMORGAN ❖ 11.1 Introduction In Class XI, while studying Analytical Geometry in two dimensions, and the introduction to three dimensional geometry, we confined to the Cartesian methods only. In the previous chapter of this book, we have studied some basic concepts of vectors. We will now use vector algebra to three dimensional geometry. The purpose of this approach to 3-dimensional geometry is that it makes the study simple and elegant*. Leonhard Euler In this chapter, we shall study the direction cosines (1707-1783) and direction ratios of a line joining two points and also discuss about the equations of lines and planes in space under different conditions, angle between two lines, two planes, a line and a plane, shortest distance between two skew lines and distance of a poin…

Practice Questions from this Chapter

Tap "Get Solution" on any question to ask our AI tutor.

1. Define direction cosines simply. Get Solution →
2. Visualize a line in space. Get Solution →
3. Calculate a 3D distance. Get Solution →
4. What are the cosines of the direction angles α, β, and γ of a directed line called? Get Solution →
5. If l, m, and n are the direction cosines of a line, what is the value of l² + m² + n²? Get Solution →
6. Any three numbers that are proportional to the direction cosines of a line are called what? Get Solution →
7. What are the direction cosines of the x-axis? Get Solution →
8. 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.

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