Question

If the vectors $\hat{i}-3\hat{j}+2\hat{k},-\hat{i}+2\hat{j}$ represents the diagonal of a parallelogram, then its area will be

• A. $\sqrt{21}$
• B. $\frac{\sqrt{21}}{2}$
• C. $2\sqrt{21}$
• D. $\frac{\sqrt{21}}{4}$

Answer 1

Answer: (B)

$\frac{\sqrt{21}}{2}$

Let $\overrightarrow{a}=\hat{i}-3\hat{j}+2\overrightarrow{k},\hat{b}=-\hat{i}+2\overrightarrow{j}$
Now, $\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -3& 2\\ -1& 2& 0\end{array}\right|$
$=-4\hat{i}-2\hat{j}-k$
Hence, required area $=\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}|$
$=\frac{1}{2}\sqrt{16+4+1}=\frac{\sqrt{21}}{2}$