Question

Find the equation of the line passing through $(1,2,-4)$ and perpendicular to both the lines $\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-4}{4}$ and $\frac{x-3}{5}=\frac{y+6}{1}=\frac{z+10}{2}$

Answer 1

Let direction ratio of new line be $(a,b,c)$

$\therefore 2a-3b+4c=0-\left(i\right)$

$5a+b+2c=0-\left(ii\right)$

Comparing equation (i) and eq. (ii), we get.

$\frac{a}{-10}=\frac{b}{16}=\frac{c}{17}$

$\therefore$ equation of line is $\frac{x-1}{-10}=\frac{y-2}{16}=\frac{z+4}{17}$