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Ex 11.1, 3 - Chapter 11 Class 12 Three Dimensional Geometry
Last updated at August 19, 2026 by Teachoo
Chapter 11 Class 12 Three Dimensional Geometry
Concept wise
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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
Transcript
Ex 11.1, 3 If a line has the direction ratios −18, 12, −4, then what are its direction cosines?If direction ratios of a line are a, b, c direction cosines are 𝒂/√(𝒂^𝟐 + 𝒃^𝟐 + 𝒄^𝟐 ) , 𝒃/√(𝒂^𝟐 + 𝒃^𝟐 + 𝒄^𝟐 ) , 𝒄/√(𝒂^𝟐 + 𝒃^𝟐 + 𝒄^𝟐 ) Given, Direction ratios = −18, 12, −4 𝒂 = −18, b = 12, c = −4 And, √(𝒂𝟐+𝒃𝟐+𝒄𝟐) = √((−18)2+122+(−4)2) √(𝒂𝟐+𝒃𝟐+𝒄𝟐) = √((−18)2+122+(−4)2) = √(324+144+16) = √484 = 22 Therefore, Direction cosines = 𝑎/√(𝑎^2 + 𝑏^2 + 𝑐^2 ) , 𝑏/√(𝑎^2 + 𝑏^2 + 𝑐^2 ) , 𝑐/√(𝑎^2 + 𝑏^2 + 𝑐^2 ) = (−18)/22 , 12/22 , (−4)/22 = (−𝟗)/𝟏𝟏 , 𝟔/𝟏𝟏 , (−𝟐)/𝟏𝟏
