Question

Show that the following points are coplanar:
$(0,-1,0),(2,1,-1),(1,1,1)$ and $(3,3,0)$

Answer 1

Let the given points are

$A\left(0,-1,0\right),B\left(2,1,-1\right),C\left(1,1,1\right),D\left(3,3,0\right)$

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

$\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=\hat{i}+2\hat{j}+\hat{k}$

$\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=3\hat{i}+4\hat{j}+0\hat{k}$

$\left[\begin{array}{ccc}\overrightarrow{AB}& \overrightarrow{AC}& \overrightarrow{AD}\end{array}\right]=\left|\begin{array}{ccc}2& 2& -1\\ 1& 2& 1\\ 3& 4& 0\end{array}\right|$

$=2\left(0-4\right)-2\left(0-3\right)-1\left(4-6\right)=0$

Therefore points $A,B,C$ and $D$ are coplanar.