Question

Find $x$ such that the four points $A(4,1,2),B(5,x,6),C(5,1,-1)$ and $D(7,4,0)$ are coplanar.

Answer 1

$A(4,1,2),B(5,x,6),C(5,1,-1),D(7,4,0)$

$AB=(1,x-1,4)$

$AC=(1,0,-3)$

$AD=(3,3,-2)$

$\Rightarrow \overrightarrow{AD}.(\overrightarrow{AB}\times \overrightarrow{AC)}=\left|\begin{array}{ccc}1& x-1& 4\\ 1& 0& -3\\ 3& 3& -2\end{array}\right|=0$

$\Rightarrow 9-7(x-1)+12=0$

$\Rightarrow 7x=28$

$\Rightarrow x=4$

Hence the answer is $4.$