Question

Find the area of the parallelogram whose diagonals are represented by the vectors $2\hat{i}+3\hat{j}-6\hat{k}$ and $3\hat{i}-4\hat{j}-\hat{k}$.

Answer 1

Diagonals of the parallelogram are represented by $2\hat{i}+3\hat{j}-6\hat{k}$ and $3i-4j-k$.

Area of parallelogram $=\frac{1}{2}|\overrightarrow{A}\times \overrightarrow{B}|$

(Here, $\overrightarrow{A},\overrightarrow{B}$ are vectors of diagonals of parallelogram).

Area $=\frac{1}{2}\left|\right(2i+3j-6k)\times (3i-4j-k\left)\right|$

$=\frac{1}{2}|-27i-16j-17k|$

$=\frac{1}{2}\sqrt{(27{)}^2+(16{)}^2+(17{)}^2}$

$=\frac{1}{2}\sqrt{1274}$

$=\frac{1}{2}7\sqrt{26}=\frac{7\sqrt{26}}{2}$.