Let A,B,C and D be the given points. Then, −−→AB=(^i+2^j+3^k)−(3^i+6^j+9^k)=−2^i−4^j−6^k−−→AC=(2^i+3^j+^k)−(3^i+6^j+9^k)=−^i−3^j−8^k−−→AD=(4^i+6^j+λ^k)−(3^i+6^j+9^k)=^i+0^j+(λ−9)^k
Given points are coplanar if vectors
−−→AB,−−→AC,−−→AD are coplanar
[−−→AB,−−→AC,−−→AD]=0⇒−2(λ−9)(−3)+4(−λ+9+8)−6(3)=0⇒6λ−54−4λ+68−18=0⇒2λ−4=0⇒λ=2