Question

If $\overrightarrow{A}=\hat{i}+2\hat{j}-\hat{k}$ and $\overrightarrow{B}=2\hat{i}+3\hat{j}+4\hat{k}$, Then the vector which is perpendicular to both vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is

• A. $11\hat{i}+6\hat{j}-\hat{k}$
• B. $5\hat{i}-6\hat{j}+\hat{k}$
• C. $11\hat{i}-6\hat{j}-\hat{k}$
• D. $5\hat{i}-6\hat{j}-\hat{k}$

Answer 1

The correct option is C $11\hat{i}-6\hat{j}-\hat{k}$

As we know that $\overrightarrow{A}\times \overrightarrow{B}$ will be the perpendicular to both vectors $\overrightarrow{A}$ and $\overrightarrow{B}$
So, we have
$\overrightarrow{A}\times \overrightarrow{B}=\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& -1\\ 2& 3& 4\end{array}\right]$
=[$4\times 2-(3\times (-1\left)\right]\hat{i}-[1\times 4-(-1\times 2\left)\right]\hat{j}+[3\times 1-2\times 2]\hat{k}$
=$11\hat{i}-6\hat{j}-\hat{k}$