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

$\overrightarrow{AB}=\hat{i}+(x-1)\hat{j}+4\hat{k}$

$\overrightarrow{BC}=0\hat{i}+(1-x)\hat{j}-7\hat{k}$

$\overrightarrow{CD}=2\hat{i}+3\hat{j}+\hat{k}$

Since the points A, B, C, D are coplanar

$\Rightarrow \left[\overrightarrow{AB}\overrightarrow{BC}\overrightarrow{CD}\right]=0$

$\Rightarrow \left|\begin{array}{ccc}1& x-1& 4\\ 0& 1-x& -7\\ 2& 3& 1\end{array}\right|=0$

$\Rightarrow 1(1-x+21)+2(-7x+7-4+4x)=0$

$\Rightarrow (22-x)+2(3-3x)=0$

$\Rightarrow 22-x+6-6x=0$

$\Rightarrow -7x=-28$

$\Rightarrow x=4$