Question

Which one of the following is the unit vector perpendicular to both $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ ?

• A. $\frac{\hat{i} + \hat{j}}{\sqrt{2}}$
• B. $\hat{k}$
• C. $\frac{\hat{j} + \hat{k}}{\sqrt{2}}$
• D. $\frac{\hat{i} - \hat{j}}{\sqrt{2}}$

Answer 1

The Correct Option is A

According to question $a=-\hat{i}+\hat{j}+\hat{k}$ and $b=\hat{i}-\hat{j}+\hat{k}$
Then , $a \times b = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ -1&1&1\\ 1&-1&1\end{vmatrix}$
$=\hat{i}[1+1]-\hat{j}[-1-1]+\hat{k}[1-1]=2(\hat{i}+\hat{j})$
and $|a \times b|=\sqrt{4+4}=2 \sqrt{2}$
$\therefore$ Required unit vector
$=\pm \frac{2(\hat{i}+\hat{j})}{2 \sqrt{2}}$
$=\pm \frac{\hat{i}+\hat{j}}{\sqrt{2}}$