Supplementary Exercise Q5
Last updated at May 29, 2018 by Teachoo
Last updated at May 29, 2018 by Teachoo
Transcript
Supplementary Exercise Q5 Find the value of 𝜆 if the following vectors are co-planar (i) 𝑎 = 2 𝑖 − 𝑗 + 𝑘, 𝑏 = 𝑖 + 2 𝑗 − 3 𝑘, and 𝑐 = 3 𝑖 − 𝜆 𝑗 + 5 𝑘 Three vectors 𝑎, 𝑏, 𝑐 are coplanar if 𝒂 𝒃 𝒄 = 0 Given, 𝑎 = 2 𝑖 − 𝑗 + 𝑘 𝑏 = 𝑖 + 2 𝑗 − 3 𝑘 𝑐 = 3 𝑖 − λ 𝑗 + 5 𝑘 𝑎 𝑏 𝑐 = 2−1112−33−𝜆5 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 = 287 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 = 2 𝑖 + 𝑗 + 𝑘, 𝑏 = 2 𝑖 − 𝜆 𝑗 + 𝑘, and 𝑐 = 5 𝑖 − 𝑗 − 3 𝑘 Three vectors 𝑎, 𝑏, 𝑐 are coplanar if 𝒂 𝒃 𝒄 = 0 Given, 𝑎 = 2 𝑖 + 𝑗 + 𝑘 𝑏 = 2 𝑖 − λ 𝑗 + 𝑘 𝑐 = 5 𝑖 + 𝑗 – 3 𝑘 𝑎 𝑏 𝑐 = 2112−𝜆151−3 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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