Question

The unit vector perpendicular to each of the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $3\hat{i}+4\hat{j}$ is

• A. $\frac{1}{\sqrt{146}}(4\hat{i}-3\hat{j}+11\hat{k})$
• B. $\frac{1}{\sqrt{146}}(-4\hat{i}+3\hat{j}+11\hat{k})$
• C. $\frac{1}{\sqrt{146}}(4\hat{i}+3\hat{j}+11\hat{k})$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{\sqrt{146}}(-4\hat{i}+3\hat{j}+11\hat{k})$

$(2\hat{i}-\hat{j}+\hat{k})\times (3\hat{i}+4\hat{j})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -1& 1\\ 3& 4& 0\end{array}\right|$

$=\hat{i}(-4)+\hat{j}\left(3\right)+\hat{k}\left(11\right)$

$\therefore$ Required unit vector

$=\frac{-4\hat{i}+3\hat{j}+11\hat{k}}{\sqrt{{(-4)}^2+3^2+{11}^2}}=\frac{-4\hat{i}+3\hat{j}+11\hat{k}}{\sqrt{146}}$