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

Options

A. 4
B. 5
C. 6
D. 7

Answer

D

Explanation

The line passes through `A(6,7,7)` and is parallel to the vector `vecb=3hati+2hatj-2hatk`
Let `M` be the foot of the perpendicular from `P(1,2,3)` on the given line.
We have
`vec(AP)=-5hati-5hatj-4hatkimplies|vec(AP)|=sqrt(25+25+16)=sqrt(66)`
Clearly, `AM=` Projection of `vec(AP)` on `vecb`
`impliesAM=|(vec(AP).vecb)/(|vecb|)|`

`AM=((5hati+5hatj+4hatk).(3hati+2hatj-2hatk))/(|3hati+2hatj-2hatk|)=(15+10-8)/(sqrt(9+4+4))=sqrt(17)`
`:.PM=sqrt(AP^(2)-AM^(2))impliesPM=sqrt(66-17)=7`