Question

Show that the four points $A(4,5,1),B(0,-1,-1),C(3,9,4),D(-4,4,4)$ are coplanar.

Answer 1

Let the position vectors of $A,B,C,D$ are :

$\overrightarrow{OA}=4\hat{i}+5\hat{j}+\hat{k},\overrightarrow{OB}=-\hat{j}-\hat{k},\overrightarrow{OC}=3\hat{i}+9\hat{j}+4\hat{k},\overrightarrow{OD}=-4\hat{i}+4\hat{j}+4\hat{k}$

Now, $\overrightarrow{AB}=-4\hat{i}-6\hat{j}-2\hat{k}$

$\overrightarrow{AC}=-\hat{i}+4\hat{j}+3\hat{k}$

$\overrightarrow{AD}=-8\hat{i}-\hat{j}+3\hat{k}$

Now, $\left[\begin{array}{ccc}\overrightarrow{AB}& \overrightarrow{AC}& \overrightarrow{AD}\end{array}\right]=\left|\begin{array}{ccc}-4& -6& -2\\ -1& 4& 3\\ -8& -1& 3\end{array}\right|$

$=-4(12+3)+6(-3+24)-2(1+32)=-60+126-66=0$

Hence, the four points $A,B,C,D$ are coplanar.