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