Question

A unit vector perpendicular to both the vectors $\hat {i} +\hat { j} $ and $ \hat {j} +\hat {k}$ is

• A. $\frac {-\hat {i}-\hat{j}+\hat {k}} {\sqrt {3}}$
• B. $\frac {\hat {i}+\hat{j}-\hat {k}} { {3}}$
• C. $\frac {\hat {i}+\hat{j}+\hat {k}} {\sqrt {3}}$
• D. $\frac {\hat {i}-\hat{j}+\hat {k}} {\sqrt {3}}$

Answer 1

Answer: (D)

Let $\vec{a} = \hat{i} + \hat{j} $ and $ \vec{b} = \hat{j} +\hat{k} $
Now, $ \vec{a} \times\vec{b} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 1&1&0\\ 0&1&1\end{vmatrix} $
$=\hat{i} \left(1-0\right) -\hat{j} \left(1-0\right) + \hat{k} \left(1-0\right)$
$ =\hat{i} -\hat{j} +\hat{k} $
and $ \left|\vec{a} \times\vec{b}\right| = \sqrt{1^{2} +\left(-1\right)^{2} +1^{2}} = \sqrt{3} $
$\therefore $ Required unit vector $ = \frac{\vec{a} \times\vec{b}}{\left|\vec{a} \times\vec{b}\right|} $
$ = \frac{\hat{i} - \hat{j} + \hat{k}}{\sqrt{3}} $