Question

Area of a rectangle having vertices $A,B,C$ and $D$ with position vectors $-\hat{i}+\frac{1}{2}\hat{j}+4\hat{k},\hat{i}+\frac{1}{2}\hat{j}+4\hat{k},\hat{i}-\frac{1}{2}\hat{j}+4\hat{k}\text{and}-\hat{i}-\frac{1}{2}\hat{j}+4\hat{k}$ respectively is

Answer 1

$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=(\hat{i}+\frac{1}{2}\hat{j}+4\hat{k})-(-\hat{i}+\frac{1}{2}\hat{j}+4\hat{k})=2\hat{i}\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=(-\hat{i}-\frac{1}{2}\hat{j}+4\hat{k})-(-\hat{i}+\frac{1}{2}\hat{j}+4\hat{k})=-\hat{j}$
Area of rectangle $=|\overrightarrow{AB}\times \overrightarrow{AD}|=|\left(2\hat{i}\right)\times (-\hat{j})|=|-2\hat{k}|=\sqrt{(-2{)}^2}=2$