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Ex 11.1, 4 - Chapter 11 Class 12 Three Dimensional Geometry
Last updated at August 13, 2026 by Teachoo
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
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Transcript
Ex 11.1, 4 Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear. B(–1, –2, 1) A(2, 3, 4) C(5, 8, 7) Three points A, B, C are collinear if direction ratios of AB and BC are proportional. AB A (2, 3, 4) B (–1, –2, 1) Direction ratios = –1 − 2, –2 −3, 1 − 4 = –3, −5, –3 So, 𝑎1, = −3 , b1 = −5, c1 = –3 BC B (–1, –2, 1) C (5, 8, 7) Direction ratios = 5 − (–1), 8 − (−2), 7 –1 = 6, 10, 6 So, 𝑎2, = 6 , b2 = 10, c2 = 6 Now, 𝒂𝟐/𝒂𝟏 = 6/( −3) = –2 𝒃𝟐/𝒃𝟏 = 10/( −5) = –2 𝒄𝟐/𝒄𝟏 = 6/(−3) = –2 Since, 𝒂𝟐/𝒂𝟏 = 𝒃𝟐/𝒃𝟏 = 𝒄𝟐/𝒄𝟏 = −2 Since direction ratios of AB and BC are proportional ∴ Points A, B and C are collinear.
