Yacht A has initial position (-10, 4) and has velocity vector.

Yacht B has initial position (3, -13) and has velocity vector.

1. Explain why the position of each yacht at time t is given by

rA = + t and rB = + t

- For vector equations, the form is =. ( refers to an initial position and refers to a direction vector.)

- Therefore, a vector equation for Yacht A can be written as + t.

- A vector equation for Yacht B can be written as + t.

3. The position of B relative to A () is

rB – rA = + t, which in coordinates will be.

4. The formula for finding the distance is:

d =

Therefore, d2 = 169 - 78t + 9t2 + 289 - 136t + 16t2

= 25t2 - 214t + 458

5. d2 is a minimum when t = 4.28

To find the minimum of d2, I set the derivative equal to 0.

So, 50t - 214 = 0. Thus t = = 4.28.

6. The time when d is to be a minimum is the same time as when d2 is a minimum, so the closest approach occurs at t = 4.28. So, if I put t = 4.28 into the expression for d is:

d =

=

=

=

= 0.2 miles