Let A ( 2, 3, 5 ) and C ( - 3, 4, - 2 ) be opposite vertices of a...

Area of the parallelogram is given by $\dfrac{1}{2}\left|\overset{\to}{AC} \times \overset{\to}{BD}\right|$.

⇒ Area = \dfrac{1}{2}\left|\left(- 5\overset{^}{i} + \overset{^}{j} - 7\overset{^}{k}\right) \times \left(\overset{^}{i} + 2\overset{^}{j} + 3\overset{^}{k}\right)\right|

⇒ Area = \dfrac{1}{2}\left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ - 5 & 1 & - 7 \\ 1 & 2 & 3\end{cases}\right|

⇒ Area = \dfrac{1}{2}\left|17\overset{^}{i} + 8\overset{^}{j} - 11\overset{^}{k}\right|

$⇒ Area = \dfrac{1}{2}\sqrt{289 + 64 + 121}$

$⇒ Area = \dfrac{\sqrt{474}}{2}$ square units.