Question

The area of the parallelogram whose adjacent sides area $\hat{i}+\hat{k}$ and $2\hat{i}+\hat{j}+\hat{k}$ is.

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. 3
• D. 4

Answer 1

The correct option is B $\sqrt{3}$

Fact: Area of parallelogram whose adjacent sides are $\overrightarrow{a}$ and $\overrightarrow{b}$

is given by $|\overrightarrow{a}\times \overrightarrow{b}|$

Here $\overrightarrow{a}=\hat{i}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{j}+\hat{k}$

So $\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 0& 1\\ 2& 1& 1\end{array}\right|=\hat{i}(0-1)-\hat{j}(1-2)+\hat{k}(1-0)=-\hat{i}+\hat{j}+\hat{k}$

Hence are of parallelogram $=|-\hat{i}+\hat{j}+\hat{k}|=\sqrt{(-{)}^2+1^2+1^2}=\sqrt{3}$