Question

The unit vector normal to the plane containing $\overrightarrow{a}=\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}$ is

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

Answer 1

The correct option is C $\frac{\hat{k}-\hat{j}}{\sqrt{2}}$

Normal unit vector to plane $=\hat{n}$
$\hat{n}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{|\overrightarrow{a}\times \overrightarrow{b}|}$
$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -1& -1\\ 1& 1& 1\end{array}\right|=\left(\right(0\hat{j})+(-2)\hat{j}+2\hat{k})$
$|\overrightarrow{a}\times \overrightarrow{b}|=2\sqrt{2}$
$\hat{n}=\frac{1}{\sqrt{2}}(\hat{k}-\hat{j})$