Question

Get the area of $△ABC$ for $A(1,1,2),B(2,3,5),C(1,3,4)$ by use of vectors.

Answer 1

$\begin{array}{c}Areaof\Delta ABC=\frac{1}{2}\left|\overrightarrow{AB}\times \overrightarrow{AC}\right|\\ \overrightarrow{AB}=\left(1,2,3\right)\\ \overrightarrow{AB}\times \overrightarrow{AC}\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 3\\ 0& 2& 2\end{array}\right|=\hat{i}\left(-2\right)-\hat{j}\left(2\right)+\hat{k}\left(2-0\right)\\ =-2\hat{i}-2\hat{j}+2\hat{k}\\ Area=\frac{1}{2}\sqrt{4+4+4}\\ =\frac{\sqrt{12}}{2}\\ =\frac{\sqrt{12}}{2}\\ =\frac{2\sqrt{3}}{2}=\sqrt{3}.\end{array}$