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Question
The vectors λ^i+^j+2^k,^i+λ^j−^k and 2^i−^j+λ^k are coplanar, if
(a) λ=−2 (b) λ=0
(a) λ=1 (b) λ=−1
The vectors λ^i+^j+2^k,^i+λ^j−^k and 2^i−^j+λ^k are coplanar, if
(a) λ=−2 (b) λ=0
(a) λ=1 (b) λ=−1
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Solution
(a) Let →a=λ^i+^j+2^k,^i+λ^j−^k and 2^i−^j+λ^k
For→a,→b and→c to be coplanar,∣∣
∣∣λ121λ−12−1λ∣∣
∣∣=0⇒ λ(λ2−1)−1(λ+2)+2(−1−2λ)=0⇒ λ2−λ−λ−2−2−4λ=0⇒ λ3−6λ−4=0⇒ (λ+2)(λ2−2λ−2)=0⇒ λ=−2 or λ=2±√122⇒ λ=−2 or λ=2±2√32=1±√3
So, option (a) is the required correct answer.
(a) Let →a=λ^i+^j+2^k,^i+λ^j−^k and 2^i−^j+λ^k
For→a,→b and→c to be coplanar,∣∣
∣∣λ121λ−12−1λ∣∣
∣∣=0⇒ λ(λ2−1)−1(λ+2)+2(−1−2λ)=0⇒ λ2−λ−λ−2−2−4λ=0⇒ λ3−6λ−4=0⇒ (λ+2)(λ2−2λ−2)=0⇒ λ=−2 or λ=2±√122⇒ λ=−2 or λ=2±2√32=1±√3
So, option (a) is the required correct answer.
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