Slide20.JPG

Slide21.JPG
Slide22.JPG
Slide23.JPG

Go Ad-free

Transcript

Example 9 Find the shortest distance between the lines l1 and l2 whose vector equations are 𝑟 ⃗ = 𝑖 ̂ + 𝑗 ̂ + 𝜆(2𝑖 ̂ − 𝑗 ̂ + 𝑘 ̂ ) and 𝑟 ⃗ = 2𝑖 ̂ + 𝑗 ̂ – 𝑘 ̂ + 𝜇 (3𝑖 ̂ – 5𝑗 ̂ + 2𝑘 ̂ )Shortest distance between lines 𝑟 ⃗ = (𝑎1) ⃗ + 𝜆 (𝑏1) ⃗ and 𝑟 ⃗ = (𝑎2) ⃗ + 𝜇(𝑏2) ⃗ is |(((𝒃𝟏) ⃗ × (𝒃𝟐) ⃗ ).((𝒂𝟐) ⃗ − (𝒂𝟏) ⃗ ))/|(𝒃𝟏) ⃗ × (𝒃𝟐) ⃗ | | 𝒓 ⃗ = (𝒊 ̂ + 𝒋 ̂) + 𝜆 (2𝒊 ̂ − 𝒋 ̂ + 𝒌 ̂) Comparing with 𝑟 ⃗ = (𝑎1) ⃗ + 𝜆 (𝑏1) ⃗ (𝒂𝟏) ⃗ = 1𝑖 ̂ + 1𝑗 ̂ + 0𝑘 ̂ & (𝒃𝟏) ⃗ = 2𝑖 ̂ – 1𝑗 ̂ + 1𝑘 ̂ 𝒓 ⃗ = (2𝒊 ̂ + 𝒋 ̂ − 𝒌 ̂) + 𝝁 (3𝒊 ̂ − 5𝒋 ̂ + 2𝒌 ̂) Comparing with 𝑟 ⃗ = (𝑎2) ⃗ + 𝜇(𝑏2) ⃗ (𝒂𝟐) ⃗ = 2𝑖 ̂ + 1𝑗 ̂ − 1𝑘 ̂ & (𝒃𝟐) ⃗ = 3𝑖 ̂ − 5𝑗 ̂ + 2𝑘 ̂ Now (𝒂𝟐) ⃗ − (𝒂𝟏) ⃗ = (2𝑖 ̂ + 1𝑗 ̂ − 1𝑘 ̂) − (1𝑖 ̂ + 1𝑗 ̂ + 0𝑘 ̂) = (2 − 1) 𝑖 ̂ + (1 − 1)𝑗 ̂ + (−1 − 0) 𝑘 ̂ = 1𝒊 ̂ + 0𝒋 ̂ − 1𝒌 ̂ (𝒃𝟏) ⃗ × (𝒃𝟐) ⃗ = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@2& −1&1@3& −5&2)| = 𝑖 ̂ [(−1×2)−(−5×1)] − 𝑗 ̂ [(2×2)−(3×1)] + 𝑘 ̂[(2×−5)−(3×−1)] = 𝑖 ̂ [−2+5] − 𝑗 ̂ [4−3] + 𝑘 ̂ [−10+3] = 𝑖 ̂ (3) − 𝑗 ̂ (1) + 𝑘 ̂(−7) = 3𝒊 ̂ − 𝒋 ̂ − 7𝒌 ̂ Magnitude of ((𝑏1) ⃗ × (𝑏2) ⃗) = √(32+(−1)2+(−7)^2 ) |(𝒃𝟏) ⃗× (𝒃𝟐) ⃗ | = √(9+1+49) = √𝟓𝟗 Also, ((𝒃𝟏) ⃗ × (𝒃𝟐) ⃗) .((𝒂𝟐) ⃗ − (𝒂𝟏) ⃗) = (3𝑖 ̂ − 𝑗 ̂ − 7𝑘 ̂) . (1𝑖 ̂ + 0𝑗 ̂ − 1𝑘 ̂) = (3 × 1) + (−1 × 0) + (−7 × −1) = 3 + 0 + 7 = 10 Therefore, Shortest distance = |(((𝑏1) ⃗ × (𝑏2) ⃗ ).((𝑎2) ⃗ − (𝑎1) ⃗ ))/|(𝑏1) ⃗ × (𝑏2) ⃗ | | = |10/√59| = 𝟏𝟎/√𝟓𝟗

Davneet Singh's photo - Co-founder, Teachoo

Made by

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.