Question

A unit vector perpendicular to both $L_1$ & $L_2$ 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

Answer: (C)

$\frac{-\hat{i}-7\hat{j}+5\hat{k}}{5\sqrt{3}}$

We have the direction vectors of the lines

$3\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$

The unit vector perpendicular to both will be in the direction of the cross product of both, i.e.

$(3\hat{i}+\hat{j}+2\hat{k})\times (\hat{i}+2\hat{j}+3\hat{k})$

$=6\hat{k}-9\hat{j}-\hat{k}+3\hat{i}+2\hat{j}-4\hat{i}$

$=-\hat{i}-7\hat{j}+5\hat{k}$

Thus the unit vector becomes

$\frac{-\hat{i}-7\hat{j}+5\hat{k}}{\sqrt{(-1{)}^2+(-7{)}^2+(5{)}^2}}$

$=\frac{-\hat{i}-7\hat{j}+5\hat{k}}{\sqrt{75}}$