Vector Equations Of Lines

Date: 6/28/96 at 1:59:19
From: Anonymous
Subject: Vector Equations Of Lines

Dr. Math, I can't solve this problem.  Can you help?

A boat leaves a harbour, O, position vector (0i+0j) at 9 a.m. 
and travels with velocity (12i+4j) km/h.  After 3 hours the 
boat develops engine problems and, with a storm forecast, the 
captain decides to head directly for shelter at harbour A, 
position vector (42i+9j) km. With the engines off the wind and current 
would cause the boat to drift with ocity (6i+2j) km/h.  With the 
faulty engine working the boat could travel at a speed of 10 km/h in 
still water.  With the engines working in this way at what velocity 
should the boat be set so that this, together with the wind and the 
current, causes the boat to arrive at A? At what time will it arrive?

Date: 6/28/96 at 9:41:38
From: Doctor Anthony
Subject: Re: Vector Equations Of Lines

If you plot the position of the boat after 3 hours (12 noon) and the 
harbour at A, you will see that the vector from the boat to A will be 
6i - 3j.

Now if the boat is driven with velocity vector ai + bj where the 
magnitude of this vector sqrt(a^2+b^2) = 10, and if t is the time from 
present position to A we have the vector equation

 t{(ai+bj) + (6i+2j)} = 6i - 3j  Now equating i and j components

  t(a+6) = 6
  t(b+2) = -3   and dividing these equations

    (a+6)/(b+2) = -2

       a+6 = -2b-4   so  a+2b = -10   also a^2 + b^2 = 100

  a = -2b-10  and a^2 = 4b^2 + 40b + 100  substitute into a^2+b^2=100

       4b^2+40b+100+b^2 = 100
             5b^2 + 40b = 0
                 b(b+8) = 0  

b=0 is not a viable result since this would require a = -10, so we 
choose the answer b = -8, and so a = -2b-10 = 16-10 = 6

And so the velocity vector of the boat in still water would be 6i - 8j

The time is given by the equation t(a+6) = 6
                                  t(6+6) = 6

                                     12t = 6  and so t = 1/2 hour

The boat therefore arrives at A at 12.30 p.m.   

-Doctor Anthony,  The Math Forum
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