Question

Find the value of $\lambda$ so that the lines.
$\frac{1-x}{3}=\frac{7y-14}{2\lambda }=\frac{z-3}{2}$ and $\frac{7-7x}{3\lambda }=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles

Answer 1

The given lines are

$\frac{x-1}{-3}=\frac{y-2}{\frac{2}{7}\lambda }=\frac{z-3}{2}$ and $\frac{x-1}{-\frac{3}{7}\lambda }=\frac{y-5}{1}=\frac{z-6}{-5}$

These lines are at right angles. Therefore,

$-3(-\frac{3}{7}\lambda )+\frac{2}{7}\lambda +2(-5)=0$

$\frac{9}{7}\lambda +\frac{2}{7}\lambda -10=0$

$\frac{11}{7}\lambda =10$

$\lambda =\frac{70}{11}$