Question

Let $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}.$
Then the magnitude of $\overrightarrow{a}\times \overrightarrow{b}$ is

[2 marks]

• A. $\sqrt{2}$
• B. $2$
• C. $2\sqrt{2}$
• D. $8$

Answer 1

The correct option is C $2\sqrt{2}$

Given : $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$
$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& -1\\ 1& -1& 1\end{array}\right|$
$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=\hat{i}(1-1)-\hat{j}(1+1)+\hat{k}(-1-1)$
$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=-2\hat{j}-2\hat{k}$
Now, magnitude of $\overrightarrow{a}\times \overrightarrow{b}$ is
$|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{(-2{)}^2+(-2{)}^2}=2\sqrt{2}$