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
\(OA=4\hat i+3\hat j+3\hat k\)
position vector of
\(OB=5\hat i+\text x\hat j+7\hat k\)
position vector of
\(OC=5\hat i+3\hat j\)
position vector of
\(OD=7\hat i+6\hat j+\hat k\)
The four points are coplanar if the vector \(\vec {AB},\vec{AC},\vec {AD}\) are coplanar
⇒ 1(9) – (x – 2)( – 2 + 9) + 4(3) = 0
⇒ 9 – 7x + 14 + 12 = 0
⇒ 35 = 7x
⇒ x = 5