Last updated at March 11, 2017 by Teachoo

Transcript

Example 12 Find the distance between the lines 𝑙1 and 𝑙2 given by 𝑟 = 𝑖 + 2 𝑗 – 4 𝑘 + 𝜆(2 𝒊 + 3 𝒋 + 6 𝒌 ) and 𝑟 = 3 𝑖 + 3 𝑗 − 5 𝑘 + μ (2 𝒊 + 3 𝒋 + 6 𝒌) Distance between two parallel lines with vector equations 𝑟 = 𝑎1 + 𝜆 𝑏 and 𝑟 = 𝑎2 + μ 𝑏 is 𝑏 × ( 𝑎2 − 𝑎1) 𝑏 Now, ( 𝒂𝟐 − 𝒂𝟏) = (3 𝑖 + 3 𝑗 − 5 𝑘) − (1 𝑖 + 2 𝑗 − 4 𝑘) = (3 − 1) 𝑖 + (3 − 2) 𝑗 + ( − 5 + 4) 𝑘 = 2 𝒊 + 1 𝒋 − 1 𝒌 Magnitude of 𝑏 = 22 + 32 + 62 𝒃 = 4+9+36 = 49 = 7 Also, 𝒃 × ( 𝒂𝟐 − 𝒂𝟏) = 𝑖 𝑗 𝑘23621−1 = 𝑖 3×−1−(1×6) − 𝑗 2×−1−(2×6) + 𝑘 2×1−(2×3) = 𝑖 −3−6 − 𝑗 −2−12 + 𝑘 2−6 = 𝑖 (–9) − 𝑗 (–14) + 𝑘(−4) = −𝟗 𝒊 + 14 𝒋 − 4 𝒌 Now, 𝒃 × ( 𝒂𝟐 − 𝒂𝟏) = −92+ 142+ −42 = 81+196+16 = 𝟐𝟗𝟑 So, distance = 𝑏 × ( 𝑎2 − 𝑎1) 𝑏 = 2937 = 𝟐𝟗𝟑𝟕 Therefore, the distance between the given two parallel lines is 2937.

Shortest distance between two parallel lines

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.