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Question

Find the value of λ for which A (3, 2, 1), B(4, 2, - 2), C (6, 5, - 1) and D (λ, 5, 5) are
coplanar.

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Solution

If A,B,C,D are coplanar vectors, then
[AB,AC,AD]=0
AB=(43)^i+(22)^j+(21)^k=^i3^k
AC=(63)^i+(52)^j+(11)^k=3^i+3^j2^k
AD=(λ3)^i+(52)^j+(51)^k=(λ3)^i+3^j+4^k
Since they are coplanar,
∣ ∣103332λ334∣ ∣
1(12+6)03(93λ+9)=0
183(183λ)=0
189(6λ)=0
26+λ=0
λ=4


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