Question

Find the area of the parallelogram three of whose vertices are $A(3,-1,2),B(1,-1,-3)$ and $C(4,-3,1)$

Answer 1

Area of 11 gm $=|\overrightarrow{AB}\times \overrightarrow{AC}|$

$=|(-2\hat{i}-5\hat{k})\times (\hat{i}-2\hat{j}-\hat{k})|$

$\Vert \begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -2& 0& -5\\ 1& -2& -1\end{array}\Vert$

$=|-10\hat{i}-7\hat{j}+4\hat{k}|$

$=\sqrt{(10{)}^2+(7{)}^2+(4{)}^2}$

$=\sqrt{165}$ square units