Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 11.2, 1 Show that the three lines with direction cosines 1213, − 313, − 413 ; 413, 1213, 313 ; 313, − 413, 1213 are mutually perpendicular. Two lines with direction cosines 𝑙1, 𝑚1 , 𝑛1 & 𝑙2, 𝑚2 , 𝑛2 are perpendicular to each other if 𝒍𝟏 𝒍𝟐 + 𝒎𝟏 𝒎𝟐 + 𝒏𝟏 𝒏𝟐 = 0 𝑙1 𝑙2 + 𝑚1 𝑚2 + 𝑛1 𝑛2 = 1213 × 413 + −313 × 1213 + −413 × 313 = 48169 + −36169 + −12169 = 48 − 36 −12169 = 48 − 48169 = 0 ∴ 𝑙1 𝑙2 + 𝑚1 𝑚2 + 𝑛1 𝑛2 = 0 Hence, the two lines are perpendicular. Now, 𝑙2 𝑙3 + 𝑚2 𝑚3 + 𝑛2 𝑛3 = 413× 313 + 1213 × − 413 + 313 × 1213 = 12169 + − 48169 + 36169 = 12 − 48 + 36169 = 48 − 48169 = 0 ∴ 𝑙2 𝑙3 + 𝑚2 𝑚3 + 𝑛2 𝑛3 = 0 Hence, the two lines are perpendicular.

Angle between two lines - Direction ratios or cosines

Chapter 11 Class 12 Three Dimensional Geometry

Concept wise

- Direction cosines and ratios
- Equation of line - given point and //vector
- Equation of line - given 2 points
- Angle between two lines - Vector
- Angle between two lines - Cartisian
- Angle between two lines - Direction ratios or cosines
- Shortest distance between two skew lines
- Shortest distance between two parallel lines
- Equation of plane - In Normal Form
- Equation of plane - Prependicular to Vector & Passing Through Point
- Equation of plane - Passing Through 3 Non Collinear Points
- Equation of plane - Intercept Form
- Equation of plane - Passing Through Intersection Of Planes
- Coplanarity of 2 lines
- Angle between two planes
- Distance of point from plane
- Angle between Line and Plane
- Equation of line under planes condition
- Point with Lines and Planes

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.