Question

Two adjacent sides of a parallelogram $ABCD$ are $2\hat{i}+4\hat{j}-5\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$, then the value of $|\overrightarrow{AC}\times \overrightarrow{BD}|$ is

• A. $20\sqrt{5}$
• B. $22\sqrt{5}$
• C. $24\sqrt{5}$
• D. $26\sqrt{5}$

Answer 1

Answer: (B)

$22\sqrt{5}$

$|\overrightarrow{AC}\times \overrightarrow{BD}|=2|\overrightarrow{AB}\times \overrightarrow{AD}|$

$=2\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 4& -5\\ 1& 2& 3\end{array}\right|$

$=|2\left[\hat{i}\right(12+10)-\hat{j}(6+5)+\hat{k}(4-4\left)\right]|$
$=|2[22\hat{i}-11\hat{j}]|$
$=22|[2\hat{i}-\hat{j}]|$
$=22\times \sqrt{5}$