Question

Find the length of perpendicular from the point $(3,2,1)$ to the line $\frac{x-7}{-2}=\frac{y-7}{2}=\frac{z-6}{3}$.

Answer 1

The given line is

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

$x=-2\lambda +7,y=2\lambda +7,z=3\lambda +6$

Direction ratios of line perpendicular to the given line are :

$-2\lambda +4,2\lambda +5,3\lambda +5$

Since, both the lines are perpendicular :

$3(-2\lambda +4)+2(2\lambda +5)+(3\lambda +5)=0$

$\lambda +27=0$

$\lambda =-27$

Hence,

$x=61,y=47,z=-75$
foot of perpendicular from $(3,2,1)$ to the given line will be $(61,47,-75)$

So, length of perpendicular,

$=\sqrt{(61-3{)}^2+(47-2{)}^2+(-75-1{)}^2}=\sqrt{3364+2025+5776}=\sqrt{11165}$ $units$