Calculus
posted by J on .
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, 82t), then
PQ=<3+4t, t1, 62t>
and we look for the value of t such that
PQ.<4,1,2>=0
or
<3+4t, t1, 62t>.<4,1,2>=0
⇒
4(3+4t)+(t1)2(62t)=0
Can you solve for t?