ABCD is a parallelogram with $\overrightarrow{A C}=\hat{i}-2 \hat{j}+\hat{k}$ and $\overrightarrow{B D}=-\hat{i}+2 \hat{j}-5 \hat{k}$. Area of this parallelogram is equal to:
Answer & explanation
Correct answer: option 2
Area vector of parallelogram
$=\frac{1}{2}(\overrightarrow{A C} \times \overrightarrow{B D})$
$=\frac{1}{2}\left|\begin{array}{ccc}i & j & k \\ 1 & -2 & 1 \\ -1 & 2 & -5\end{array}\right|$
$=\frac{1}{2}(8 \hat{i}+4 \hat{j})$
$=4 \hat{i}+2 \hat{j}$
∴ Area of the parallelogram = $|4 \hat{i}+2 \hat{j}|=2 \sqrt{5}$ sq. units
Hence (2) is correct answer.