Question

Show that the points $A(2,1,-1),B(0,-1,0),C(4,0,4)$ and $D(2,0,1)$ are coplanar.

Answer 1

Given: $A(2,1,-1),B(0,-1,0),C(4,0,4)$ and $D(2,0,1)$

Therefore,

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

$\overrightarrow{OB}=-\hat{j}$

$\overrightarrow{OC}=4\hat{i}+4\hat{k}$ and

$\overrightarrow{OD}=2\hat{i}+\hat{k}$

Thus,

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

$\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=2\hat{i}-\hat{j}+5\hat{k}$

$\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=-\hat{j}+2\hat{k}$

Now, if $|\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}|=0$ then we say that the points are coplanar.

Therefore,

$=\left|\begin{array}{ccc}-2& -2& 1\\ 2& -1& 5\\ 0& -1& 2\end{array}\right|$

$=-2(-2+5)-(-2)(4-0)+\left(1\right)(-2-0)$

$=-6+8-2$

$=0$

Hence, the points are coplanar.