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Question
If the four points with position vectors −2^i+^j+^k, ^i+^j+^k, ^j−^k and λ^j+^k are coplanar, then λ=
If the four points with position vectors −2^i+^j+^k, ^i+^j+^k, ^j−^k and λ^j+^k are coplanar, then λ=
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Solution
The correct option is A 1
Let the four given points be
→a=−2^i+^j+^k,→b=^i+^j+^k,→c=^j−^k,→d=λ^j+^k
From the given points, we can conclude
→b−→a=3^i→c−→b=−^i−2^k→d−→c=(λ−1)^j+2^k
Since, the given points are co-planar
∴∣∣
∣∣300−10−20λ−12∣∣
∣∣=0
⇒6(1−λ)=0⇒λ=1
Let the four given points be
→a=−2^i+^j+^k,→b=^i+^j+^k,→c=^j−^k,→d=λ^j+^k
From the given points, we can conclude
→b−→a=3^i→c−→b=−^i−2^k→d−→c=(λ−1)^j+2^k
Since, the given points are co-planar
∴∣∣ ∣∣300−10−20λ−12∣∣ ∣∣=0
⇒6(1−λ)=0⇒λ=1
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