Formula: -
(iii) Three vectors \(\vec a,\,\vec b\) and \(\vec c\) are coplanar if and only if \(\vec a.(\vec b\times\vec c)=0\)
let position vector of
\(OA=3\hat i+2\hat j+\hat k\)
position vector of
\(OB=4\hat i+\lambda\hat j+5\hat k\)
position vector of
\(OC=4\hat i+2\hat j-2\hat k\)
position vector of
\(OD=6\hat i+5\hat j-\hat k\)
The four points are coplanar if the vector \(\vec{AB},\vec {AC},\vec{AD}\) are coplanar
⇒ 1(9) – (λ – 2)( – 2 + 9) + 4(3 – 0) = 0
⇒ 7λ = 35
⇒ λ = 5