Find the shortest distance from a point P(2,-1,2) to a line L r= [-1,0,7] + t [4,1,-2].
Find the shortest distance from a point P(2,-1,2) to a line L r= [-1,0,7] + t [4,1,-2].
Assume a point Q on the line such that PQ is the shortest possible distance between them.
Then PQ is orthogonal to the line L.
Given P(2,-1,2).
Let Q(-1+4t, 0+t, 8-2t), then
PQ=<-3+4t, t-1, 6-2t>
and we look for the value of t such that
PQ.<4,1,-2>=0
or
<-3+4t, t-1, 6-2t>.<4,1,-2>=0
⇒
4(-3+4t)+(t-1)-2(6-2t)=0
Can you solve for t?