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Question

The position vectors of the points A,B,C and D are 3^i2^j^k,2^i+3^j4^k,^i+^j+2^k and 4^i+5^j+λ^k respectively. If the points A,B,C and D lie on a plane, find the value of λ.

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Solution

OA=3^i2^j^kOB=2^i+3^j4^kOC=^i+^j+2^kOD=4^i+4^j+λ^kAB=OBOA=^i+5^j3^kAC=OCOA=4^i+3^j+3^kAD=ODOA=^i+7^j+(λ+1)^k

Given points will be coplanar, If

[ABACAD]=0

∣ ∣15343317λ+1∣ ∣=01(3λ+321)5(4λ43)3(283)=03λ+18+35+20λ+93=017λ+146=0λ=14617

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