Question

If $\hat{i}-\hat{j}+2\hat{k},2\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}-\hat{j}+2\hat{k}$ are position vectors of vertices of a triangle, if its area is $\sqrt{k}$ square units then find $k$ .

Answer 1

Let $\overrightarrow{O A}=\hat{i}-\hat{j}+2\hat{k},\overrightarrow{O B}=2\hat{i}+\hat{j}-\hat{k}$
and $\overrightarrow{O C }=3\hat{i}-\hat{j}+2\hat{k}$
are position vectors of vertices of a triangle.
$\overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=\hat{i}+2\hat{j}-3\hat{k}$
$\overrightarrow{A C}=\overrightarrow{O C}-\overrightarrow{O A}=2\hat{i}$
Area $=\frac{1}{2}\left|\overrightarrow{A B} \times \overrightarrow{A C}\right|$
$=\sqrt{13}$