Find the length of the perpendicular from ( 3,2,1) to the line `(x - 7) /(-2) = (y - 7)/(2) = (z - 6 ) /(3)`.

Explanation

The given equation of the line is , `(x-7)/(-2)=(y-7)/(2)=(z-6)/(3)=r` (Let)
The coordinates of any point on the line are given by
`x=-2r+7,y=2r+7,z=3r+6`
Corrdinates of intersection point P,

`(-2r+7,2r+7,3r+6)`
The direction ratio 's of AP are
`(-2r+7-3,2r+7-2,3r+6-1)`
i.e., `(-2r+4,2r+5,3r+5)`
The direction ratios of the given line are -2,2 and 3.
Since AP is perpendicular to the given line ,we get
`-2(-2r+4)+2(2r+5)+3(3r+5)=0`
`4r-8+4r+10+9r+15=0`
17 r+17=0
r=-1
The coordinate of point P is (9,5,3).
The length of the perpendicular from P to the given line ,
`APsqrt((9-3)^(2)+(5-2)^(2)+(3-1)^(2))`
`=sqrt(36+9+4)=7`