Question

The vector along $L_1$ is $3\hat{i}+\hat{j}+2\hat{k}$ and along $L_2$ is $\hat{i}+2\hat{j}+3\hat{k}$. The unit vector perpendicular to both $L_1$ and $L_2$ is

• A. $\frac{-\hat{i}+7\hat{j}+7\hat{k}}{\sqrt{99}}$
• B. $\frac{-\hat{i}-7\hat{j}+5\hat{k}}{5\sqrt{3}}$
• C. $\frac{-\hat{i}+7\hat{j}+5\hat{k}}{5\sqrt{3}}$
• D. $\frac{7\hat{i}-7\hat{j}-\hat{k}}{\sqrt{99}}$

Answer 1

The correct option is B $\frac{-\hat{i}-7\hat{j}+5\hat{k}}{5\sqrt{3}}$

Given, vector along $L_1$ is $3\hat{i}+\hat{j}+2\hat{k}$ vector along $L_2$ is $\hat{i}+2\hat{j}+3\hat{k}$

Let $\overrightarrow{r}=l\hat{i}+m\hat{j}+n\hat{k}$ be the vector perpendicular to both $L_1$ and $L_2$

$3l+m+2n=0$ ....(1)

$l+2m+3n=0$ ....(2)

Eliminating $l,m,n$ from (1) and (2), we get

$\left|\begin{array}{ccc}i& j& k\\ 3& 1& 2\\ 1& 2& 3\end{array}\right|=-i-7j+5k$

Hence, a unit vector is $\frac{-i-7j+5k}{\sqrt{1+49+25}}=\frac{-i-7j+5k}{5\sqrt{3}}$