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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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo