3
Question
If the points with position vectors −(^j+^k),4^i+5^j+λ^k,3^i+9^j+4^k and −4^i+4^j+4^k are coplanar, then the value of λ is
If the points with position vectors −(^j+^k),4^i+5^j+λ^k,3^i+9^j+4^k and −4^i+4^j+4^k are coplanar, then the value of λ is
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Solution
The correct option is C 1
Let the four points be P, Q, R, S respectively. The
−−→PQ=4^i+6^j+(λ+1)^k,−−→QR=−^i+4^j+(4−λ)^k,−−→RS=−7^i−5^j+0^k,
The given points are coplanar, therefore −−→PQ,−−→QR,−−→RS are coplanar and so [−−→PQ,−−→QR,−−→RS]=0
⇒∣∣
∣∣46λ+1−144−λ−7−50∣∣
∣∣=0⇒λ=1
Let the four points be P, Q, R, S respectively. The
−−→PQ=4^i+6^j+(λ+1)^k,−−→QR=−^i+4^j+(4−λ)^k,−−→RS=−7^i−5^j+0^k,
The given points are coplanar, therefore −−→PQ,−−→QR,−−→RS are coplanar and so [−−→PQ,−−→QR,−−→RS]=0
⇒∣∣ ∣∣46λ+1−144−λ−7−50∣∣ ∣∣=0⇒λ=1
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