Question

A vector perpendicular to both vector $\overrightarrow{A}=\hat{i}+2\hat{j}+\hat{k}$ as well as $\overrightarrow{B}=\hat{i}+\hat{j}-\hat{k}$ is

• A. $-6\hat{i}+4\hat{j}-2\hat{k}$
• B. $3\hat{i}+2\hat{j}-\hat{k}$
• C. $-6\hat{i}+4\hat{j}+2\hat{k}$
• D. $3\hat{i}+2\hat{j}+\hat{k}$

Answer 1

The correct option is A $-6\hat{i}+4\hat{j}-2\hat{k}$

$\overrightarrow{A}=\hat{i}+2\hat{j}+\hat{k}\overrightarrow{B}=\hat{i}+\hat{j}-\hat{k}$

The vector perpendicular to $2$ vectors is calculated with cross product.

$$\left[\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ 1& 2& 1\\ 1& 1& -1\end{array}\right]$$ and $⟨g,h,i⟩$ are $2$ vectors Here we have $\overrightarrow{a}=⟨1,2,1⟩\overrightarrow{b}=⟨1,1,-1⟩$

Therefore

$=\overrightarrow{i}\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right]-\overrightarrow{j}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]+\overrightarrow{k}\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]=\overrightarrow{i}(-2-1)-\overrightarrow{j}(-1-1)+\overrightarrow{k}(1-2)\overrightarrow{c}=-3\hat{i}+2\hat{j}-1\hat{k}$

$\overrightarrow{2c}=-6\hat{i}+4\hat{j}-2\hat{k}$

Hence Option $A$ is correct.