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


Transcript

Supplementary Exercise Q5 Find the value of 𝜆 if the following vectors are co-planar (i) 𝑎 ⃗ = 2𝑖 ̂ − 𝑗 ̂ + 𝑘 ̂, 𝑏 ⃗ = 𝑖 ̂ + 2𝑗 ̂ − 3𝑘 ̂, and 𝑐 ⃗ = 3𝑖 ̂ − 𝜆𝑗 ̂ + 5𝑘 ̂ Three vectors 𝑎 ⃗, 𝑏 ⃗, 𝑐 ⃗ are coplanar if [𝒂 ⃗" " 𝒃 ⃗" " 𝒄 ⃗ ] = 0 Given, 𝑎 ⃗ = 2𝑖 ̂ − 𝑗 ̂ + 𝑘 ̂ 𝑏 ⃗ = 𝑖 ̂ + 2𝑗 ̂ − 3𝑘 ̂ 𝑐 ⃗ = 3𝑖 ̂ − λ 𝑗 ̂ + 5𝑘 ̂ [𝑎 ⃗" " 𝑏 ⃗" " 𝑐 ⃗ ] = |■8(2&−1&1@1&2&−3@3&−𝜆&5)| 0 = 2[(2×5)−(−λ×−3) ] − (−1) [(1×5)−(3×−3) ] + 1[(1×−λ)−(3×2) ] 0 = 2[10−3λ] + [5+9 ] + [−λ−6] 0 = 20 – 6λ + 14 – λ +– 6 0 = –7λ + 28 7λ = 28 λ = 28/7 𝝀 = 4 Therefore, 𝑎 ⃗,𝑏,𝑐 ⃗ are coplanar if λ = 4 Supplementary Exercise Q5 Find the value of 𝜆 if the following vectors are co-planar (ii) 𝑎 ⃗ = 2𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂, 𝑏 ⃗ = 2𝑖 ̂ − 𝜆𝑗 ̂ + 𝑘 ̂, and 𝑐 ⃗ = 5𝑖 ̂ − 𝑗 ̂ − 3𝑘 ̂ Three vectors 𝑎 ⃗, 𝑏 ⃗, 𝑐 ⃗ are coplanar if [𝒂 ⃗" " 𝒃 ⃗" " 𝒄 ⃗ ] = 0 Given, 𝑎 ⃗ = 2𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂ 𝑏 ⃗ = 2𝑖 ̂ − λ𝑗 ̂ + 𝑘 ̂ 𝑐 ⃗ = 5𝑖 ̂ + 𝑗 ̂ – 3𝑘 ̂ [𝑎 ⃗" " 𝑏 ⃗" " 𝑐 ⃗ ] = |■8(2&1&1@2&−𝜆&1@5&1&−3)| 0 = 2[(−λ×−3)−(1×1)] − 1[(2×−3)−(5×1)] + 1[(2×1)−(5×−λ) ] 0 = 2[3λ−1] − 1[−6−5] + 1[2+5λ] 0 = 6λ – 2 + 11 + 2 + 5λ 0 = 11λ + 11 –11λ = 11 λ = 11/(−11) 𝝀 = –1 Therefore, 𝑎 ⃗,𝑏,𝑐 ⃗ are coplanar if λ = –1

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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.