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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.

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Solution

Let A,B,C and D be the given points. Then, AB=(^i+2^j+3^k)(3^i+6^j+9^k)=2^i4^j6^kAC=(2^i+3^j+^k)(3^i+6^j+9^k)=^i3^j8^kAD=(4^i+6^j+λ^k)(3^i+6^j+9^k)=^i+0^j+(λ9)^k
Given points are coplanar if vectors
AB,AC,AD are coplanar
[AB,AC,AD]=02(λ9)(3)+4(λ+9+8)6(3)=06λ544λ+6818=02λ4=0λ=2

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