Question

Show that the four points $(0,-1,-1),(4,5,1),(3,9,4)$ and $(-4,4,4)$ are coplanar.

Answer 1

Given four points $(0,-1,-1),(4,5,1),(3,9,4)$ and $(-4,4,4)$

$\therefore$

$\overrightarrow{OA}=-\hat{j}-\hat{k}$

$\overrightarrow{OB}=4\hat{i}+5\hat{j}+\hat{k}$

$\overrightarrow{OC}=3\hat{i}+9\hat{j}+4\hat{k}$

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

$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=4\hat{i}+5\hat{j}+\hat{k}+\hat{j}+\hat{k}=4\hat{i}+6\hat{j}+2\hat{k}$

$\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=3\hat{i}+9\hat{j}+4\hat{k}+\hat{j}+\hat{k}=3\hat{i}+10\hat{j}+5\hat{k}$

$\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=-4\hat{i}+4\hat{j}+4\hat{k}+\hat{j}+\hat{k}=-4\hat{i}+4\hat{j}+4\hat{k}+\hat{j}+\hat{k}=-4\hat{i}+5\hat{j}+5\hat{k}$

$\therefore |\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}|=\left|\begin{array}{ccc}4& 6& 2\\ 3& 10& 5\\ -4& 5& 5\end{array}\right|$

$=4(50-25)-6(15+20)+2(15+40)$

$=100-210+110=0$

Hence the given points are coplanar.