Question

Find the area of a parallelogram whose adjacent sides are given by the vectors $\overrightarrow{a}=3\hat{i}+\hat{j}+4\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$.

Answer 1

The adjacent sides of parallelogram are given by vectors $a=3\hat{i}+\hat{j}+4\hat{k}$ and $b=\hat{i}-\hat{j}+\hat{k}$

We know that, if $\overrightarrow{a}$ and $\overrightarrow{b}$ are adjacent side vectors of parallelogram then the area of parallelogram is given by $|a\times b|$

So the area of parallelogram is $A=|\overrightarrow{a}\times \overrightarrow{b}|=|(3\hat{i}+\hat{j}+4\hat{k})\times (\hat{i}-\hat{j}+\hat{k})|$

$⟹A=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 1& 4\\ 1& -1& 1\end{array}\right|=|\hat{i}(1+4)-\hat{j}(3-4)+\hat{k}(-3-1)|=|5\hat{i}+\hat{j}-4\hat{k}|=\sqrt{42}$

Therefore the area of parallelogram is $\sqrt{42}$.