Examples

Chapter 11 Class 12 Three Dimensional Geometry
Serial order wise

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### Transcript

Example 8 Find the angle between the pair of lines (π₯ + 3)/3 = (π¦ β 1)/5 = (π§ + 3)/4 and (π₯ + 1)/1 = (π¦ β 4)/1 = (π§ β 5)/2Angle between the pair of lines (π₯ β π₯1)/π1 = (π¦ β π¦1)/π1 = (π§ β π§1)/π1 and (π₯ β π₯2)/π2 = (π¦ β π¦2)/π2 = (π§ β π§2)/π2 is given by cos ΞΈ = |(π_π π_π + π_π π_π +γ πγ_π π_π)/(β(γπ_πγ^π + γπ_πγ^π+ γπ_πγ^π ) β(γπ_πγ^π +γγ πγ_πγ^π+ γπ_πγ^π ))| (π + π)/π = (π β π)/π = (π + π)/π (π₯ β (β3))/3 = (π¦ β 1)/5 = (π§ β (β3))/4 Comparing with (π₯ β π₯1)/π1 = (π¦ β π¦1)/π1 = (π§ β π§1)/π1 x1 = β3, y1 = 1, z1 = β3 & π1 = 3, b1 = 5, c1 = 4 (π + π)/π = (π β π)/π = (π β π)/π (π₯ β (β1))/1 = (π¦ β 4)/1 = (π§ β 5)/2 Comparing with (π₯ β π₯2)/π2 = (π¦ β π¦2)/π2 = (π§ β π§2)/π2 π₯2 = β1, y2 = 4, z2 = 5 & π2 = 1, π2 = 1, π2 = 2 Now, cos ΞΈ = |(π_1 π_2 + π_1 π_2 +γ πγ_1 π_2)/(β(γπ_1γ^2 + γπ_1γ^2+ γπ_1γ^2 ) β(γπ_2γ^2 +γγ πγ_2γ^2+ γπ_2γ^2 ))| = |((3 Γ 1) + (5 Γ 1) + (4 Γ 2))/(β(3^2 + 5^2 + 4^2 ) Γ β(1^2 + 1^2 + 2^2 ))| = |(3 + 5 + 8)/(β(9 + 25 + 16) β(1 + 1 + 4))| = |16/(β50 β6)| = |16/(5β2 Γ β2 β3)| = |16/(5 Γ 2 Γ β3)| = 8/(5 β3) = 8/(5 β3) Γ β3/β3 = (8β3)/(15 ) So, cos ΞΈ = (8β3)/(15 ) β΄ ΞΈ = cos-1((πβπ)/(ππ )) Therefore, the angle between the given pair of line is cosβ1 ((8β3)/(15 ))

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.