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Finding the unknown cordinates of a point on a vector

  1. Dec 18, 2012 #1
    Relative to the origin O, the position vectors of two points A and B are (1,4) and (7,1) respectively. Give that the point P (t,t+1) is on AB find

    1) AP and BP in terms of t

    2) Find the value of t and hence the ratio AP:PB


    Solution:

    1) AP= (-i - 4 j) + ti + (t+1)J = (t-1)i + (t-3)j
    BP= -7i -j + ti + (t+1)j =(t-7)i + (t)j

    2) I have no idea how to find t.
     
    Last edited: Dec 18, 2012
  2. jcsd
  3. Dec 18, 2012 #2

    Dick

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    AP and PB are parallel vectors, since P is on AB, right? How do you express two vectors being parallel in algebra? BTW your expression for AP has a sign mistake.
     
    Last edited: Dec 18, 2012
  4. Dec 18, 2012 #3
    Well if AP and PB are parallel then their cross product is 0.

    0 = AP X PB

    0= ((t -1)i + (t-3)j) X (( T-7)i + tj)
    0 = - (t-3)(t-7)k + (t-1)t k
    0 = (3-t)(t-7) k + (t-1)tk
    0= (3t -t^2-21+7t)k + (t-1)tk
    0= (10t -t^2 -21)k + (t-1)tk
    0= (10t -t^2 -21)k +(t^2 - t)k
    0= (9t -21)k

    9t-21=0
    9t=21
    t= 7/3
     
    Last edited: Dec 18, 2012
  5. Dec 18, 2012 #4

    Dick

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    That would be one way to go if you extend them to three dimensional vectors. It's also true that if they are parallel then they are multiples of each other. k*AP=PB for some constant k.
     
  6. Dec 18, 2012 #5

    Wouldn't that just introduce another unknown variable which would require us to have 2 equations in order to solve. I thought this route already but didn't bother going this way since I don't know the ratio, and hence the value of k.
     
    Last edited: Dec 18, 2012
  7. Dec 18, 2012 #6

    Dick

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    You are being pretty sloppy here. (t-7)i + (t)j turned into (( T-7)i + j). Something missing. If you do this right the t^2 will cancel.
     
  8. Dec 18, 2012 #7

    Dick

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    Once you split into components it WILL turn into two equations in the two unknowns t and k. It's about the same amount of work to do it this way as to solve the cross product equation.
     
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