M1 Finding the Minimum Distance Between Two Moving Points

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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.

r1(t)=3i+4j+t(5i-j) and r2(t)=5i-2j+t(i-3j)

In general the formula between two points in the plane, is

If we write general points on r1 and r2 in coordinate form, remembering that I and j label the and 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

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