5
Question
Find the value of λ,if four points with position vectors 3^i+6^j+9^k,^i+2^j+3^k,2^i+3^j+^k and 4^i+6^j+λ^k are coplanar.
Find the value of λ,if four points with position vectors 3^i+6^j+9^k,^i+2^j+3^k,2^i+3^j+^k and 4^i+6^j+λ^k are coplanar.
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Solution
Let A(3^i+6^j+9^k),B(^i+2^j+3^k),C(2^i+3^j+^k),and D(4^i+6^j+λ^k)
∴→a=−−→OB−−−→OA=−2^i−4^j−6^k,→b=−−→OC−−−→OB=^i+^j−2^k,→c=−−→OD−−−→OC=2^i+3^j+(λ−1)^k
As the points are coplanar so,[→a →b →c]=∣∣
∣∣−2−4−611−223λ−1∣∣
∣∣=0
⇒−2(λ−1+6)+4(λ−1+4)−6(3−2)=0⇒−2λ−10+4λ+12−6=0 ∵λ=2
Let A(3^i+6^j+9^k),B(^i+2^j+3^k),C(2^i+3^j+^k),and D(4^i+6^j+λ^k)
∴→a=−−→OB−−−→OA=−2^i−4^j−6^k,→b=−−→OC−−−→OB=^i+^j−2^k,→c=−−→OD−−−→OC=2^i+3^j+(λ−1)^k
As the points are coplanar so,[→a →b →c]=∣∣ ∣∣−2−4−611−223λ−1∣∣ ∣∣=0
⇒−2(λ−1+6)+4(λ−1+4)−6(3−2)=0⇒−2λ−10+4λ+12−6=0 ∵λ=2
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