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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 7 Find the angle between the pair of lines given by 𝑟 ⃗ = 3𝑖 ̂ + 2𝑗 ̂ – 4𝑘 ̂ + 𝜆 (𝑖 ̂ + 2𝑗 ̂ + 2𝑘 ̂) and 𝑟 ⃗ = 5𝑖 ̂ – 2𝑗 ̂ + 𝜇(3𝑖 ̂ + 2𝑗 ̂ + 6𝑘 ̂)Angle between two lines 𝑟 ⃗ = (𝑎1) ⃗ + 𝜆 (𝑏1) ⃗ & 𝑟 ⃗ = (𝑎2) ⃗ + 𝜇 (𝑏2) ⃗ is cos θ = |((𝒃𝟏) ⃗ . (𝒃𝟐) ⃗)/|(𝒃𝟏) ⃗ ||(𝒃𝟐) ⃗ | | 𝒓 ⃗ = (3𝒊 ̂ + 2𝒋 ̂ − 4𝒌 ̂) + 𝜆 (𝒊 ̂ + 2𝒋 ̂ + 2𝒌 ̂) So, (𝑎1) ⃗ = 3𝑖 ̂ + 2𝑗 ̂ − 4𝑘 ̂ (𝒃𝟏) ⃗ = 1𝒊 ̂ + 2𝒋 ̂ + 2𝒌 ̂ 𝒓 ⃗ = (5𝒊 ̂ – 2𝒋 ̂ + 0𝒌 ̂) + 𝝁 (3𝒊 ̂ + 2𝒋 ̂ + 6𝒌 ̂) So, (𝑎2) ⃗ = 5𝑖 ̂ − 2𝑗 ̂ + 0𝑘 ̂ (𝒃𝟐) ⃗ = 3𝒊 ̂ + 2𝒋 ̂ + 6𝒌 ̂ Now, (𝒃𝟏) ⃗ . (𝒃𝟐) ⃗ = (1𝑖 ̂ + 2𝑗 ̂ + 2𝑘 ̂). (3𝑖 ̂ + 2𝑗 ̂ + 6𝑘 ̂) = (1 × 3) + (2 × 2) + (2 × 6) = 3 + 4 + 12 = 19 Magnitude of (𝑏1) ⃗ = √(12 + 22 + 22) |(𝒃𝟏) ⃗ | = √(1 + 4 + 4) = √9 = 3 Magnitude of (𝑏2) ⃗ = √(32 + 22 + 62) |(𝒃𝟐) ⃗ | = √(9 + 4 + 36) = √49 = 7 Therefore, cos θ = |((𝑏1) ⃗.(𝑏2) ⃗)/|(𝑏1) ⃗ ||(𝑏2) ⃗ | | cos θ = |19/(3 × 7 )| cos θ = 19/21 ∴ θ = cos-1 (𝟏𝟗/𝟐𝟏) Therefore, the angle between the pair of lines is cos-1 (19/21)

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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.