Question

$A(1,1,2),B(2,3,5),C(1,3,4)andD(0,1,1)$ are the vertices of a parallelogram $ABCD$. Find its area.

Answer 1

$A(1,1,1)$ $B(2,3,5)$ $C(1,3,4)$ $D(0,1,1)$

Parallelogram ABCD

Area of parallelogram ABCD $=|\overrightarrow{AB}\times \overrightarrow{AC}|$

$\overrightarrow{AB}=\hat{i}+2\hat{j}+4\hat{k}$

$\overrightarrow{AC}=2\hat{j}+3\hat{k}$

$\overrightarrow{AB}\times \overrightarrow{AC}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 4\\ 0& 2& 3\end{array}\right|=-2\hat{i}-3\hat{j}+2\hat{k}$

$|-2\hat{i}-3\hat{j}+2\hat{k}|=$ Area

Area $=\sqrt{(-2{)}^2+(-3{)}^2+(2{)}^2}$

Area $=\sqrt{17}$ square units.