The area of the quadrilateral A B C D with vertices A ( 2 , 1 , 1 ) ,   B ( 1 , 2 , 5 ) ,   C ( -…

Given,

The quadrilateral $ABCD$ with vertices $A(2,1,1), B(1,2,5), C(-2,-3,5)$ and $D(1,-6,-7)$

Now finding,  $\overrightarrow{AB}=-\hat{i}+\hat{j}+4\hat{k}$ and $\overrightarrow{AD}=-\hat{i}-7\hat{j}-8\hat{k}$

And plotting the diagram we get,

Now we know that,

Area of $∆ABD$ is given by,

$=\frac{1}{2}\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -1& 1& 4\\ -1& -7& 8\end{array}\right|$

$=|10\hat{i}-6\hat{j}+4\hat{k}|=2\sqrt{38}$

Now finding, $\overrightarrow{CB}=3\hat{i}+5\hat{j}$ and $\overrightarrow{CD}=3\hat{i}-3\hat{j}-12\hat{k}$
So, Area of $∆CBD=\frac{1}{2}\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& -3& -12\\ 3& 5& 0\end{array}\right|$

$=\left|6\right(5\hat{i}-3\hat{j}-2\hat{k}\left)\right|$

$=6\sqrt{38}$
$\therefore$ Area of quadrilateral $ABCD=8\sqrt{38}$ square units