What is the area of the parallelogram having diagonals $ 3\bar{i}+\bar{j}-2\bar{k}\;and\;\bar{i}-3\bar{j}+4\bar{k}$?

Answer 1

Toolbox:

• Area of parallelogram =$\large\frac{1}{2}|\overrightarrow a\times\overrightarrow b|$ where $\large\overrightarrow a\:\:and\:\:\overrightarrow b$ are diagonals of the prarallelogram.
• $\large\overrightarrow a\times\overrightarrow b=\left|\begin{array}{ccc}\hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{array}\right|$

Given that $\large\overrightarrow a=3\hat i+\hat j-2\hat k$ and

$\large\overrightarrow b=\hat i-3\hat j+4\hat k$ are diagonals of the parallelogram.

We know that$\large\overrightarrow a\times\overrightarrow b=\left|\begin{array}{ccc}\hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{array}\right|$

$=\left|\begin{array}{ccc}\hat i&\hat j&\hat k\\3&1&-2\\1&-3&4\end{array}\right|$ $=\large\:\hat i(4-6)-\hat j(12-(-2))+\hat k(-9-1)$

$\large=-2\hat i-14\hat j-10\hat k$

$\large|\overrightarrow a\times\overrightarrow b|=\sqrt{(-2)^2+(-14)^2+(-10)^2}$

$=\large\sqrt{4+196+100}=\sqrt{300}=10\sqrt3$

We know that area of the parallelogram =$\large\frac{1}{2}|\overrightarrow a\times\overrightarrow b|$ where $\large\overrightarrow a\:\:and\:\:\overrightarrow b$ are diagonals of the prarallelogram.

Area of the parallelogram $\large=\frac{1}{2}10\sqrt3=5\sqrt3$