Question

Find the value of $x$ such that the points $A(3,2,1),B(4,x,5),C(4,2,-2)\text{and}D(6,5,-1)$ are coplanar.

Answer 1

Let $A,B,C$ and $D$ be the given points.Then,

$\overrightarrow{AB}=(4\hat{i}+x\hat{j}+5\hat{k})-(3\hat{i}+2\hat{j}+\hat{k})$

= $\hat{i}+(x-2)\hat{j}+4\hat{k}$ $\overrightarrow{AC}=(4\hat{i}+2\hat{j}-2\hat{k})-(3\hat{i}+2\hat{j}+\hat{k})=\hat{i}-3\hat{k}$

$\overrightarrow{AD}=(6\hat{i}+5\hat{j}-\hat{k})-(3\hat{i}+2\hat{j}+\hat{k})=3\hat{i}+3\hat{j}-2\hat{k}$

Given points are coplanar if vectors $\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}$ are coplaner

i.e, $\left[\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}\right]=0$

$\left[\begin{array}{ccc}1& x-2& 4\\ 1& 0& -3\\ 3& 3& -2\end{array}\right]=0$

$1(0+9)-(x-2)(-2+9)+4\left(3\right)=09-7x+14+12=0$

$\Rightarrow 7x=35\Rightarrow x=5$