## The Least Distance Between Moving Points or Two Lines

Suppose we have two points. Both points are moving. Up to a certain time they are moving towards each other and afterwards they are moving away from each other. We need to find the minimum distance between the points and the the value of time, t, that gives this minimum distance.

r _{1 }(t)=3i+4j+t(5i-j) and r _{2 }(t)=5i-2j+t(i-3j)

In general the formula between two points in theplane, is

If we write general points on r _{1 }and r _{2 }in coordinate form, remembering that I and j label theand coordinates respectively, we get the points:

and

And the distance between the points is

We now follow the usual procedure for completing the square.

Inside the square root, if we now put t=-0.4, the squared term is zero so the least distance is