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How would I find a vector parametric equation through these points?

  1. Oct 9, 2011 #1
    1. The problem statement, all variables and given/known data

    Find a vector parametric equation r(t) for the line through the points P=(3,0,4) and Q=(1,−3,9) for each of the given conditions on the parameter t.

    I'm stuck on this one:

    r(5)=P and r(8)=Q

    2. Relevant equations



    3. The attempt at a solution

    I tried finding the parametric equations but it didn't work.

    I don't really understand how to go about solving this. What is it asking exactly and how would I start?

    Any help is appreciated.
     
  2. jcsd
  3. Oct 9, 2011 #2

    LCKurtz

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    Can you find any parametric equation of the line through those two points? Let's start there; then we can talk.
     
  4. Oct 9, 2011 #3
    @LCKurtz: Yeah I used the old method of finding the parametric equations, taking PQ as the vector, and using either P or Q as points. That didn't work out. I actually figured out how to do it, I just want to understand it. Like geometrically, what is the question saying?
     
  5. Oct 9, 2011 #4

    LCKurtz

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    The "standard" parametric equation with that direction vector and starting at P is at the point P when t = 0 and Q when t = 1. The point of the problem is to rescale the parameter t so that it is at point P when t = 3 and Q when t = 8.
     
  6. Oct 9, 2011 #5

    HallsofIvy

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    Suppose you just start with the general linear formulas x= at+ b, y= ct+ d, z= et+ f and use the facts that
    x(5)= 3, y(5)= 0, z(5)= 4 and x(8)= 1, y(8)= −3, z(8)= 9.

    You have three sets of two equations in two unknowns.
     
  7. Oct 9, 2011 #6
    @HallsofIvy and LCKurtz: Thanks, so basically i'm re-scaling the original linear equations to fit into the new parameters? Sorry, I'm just trying to understand the geometrical aspect.
     
  8. Oct 10, 2011 #7

    HallsofIvy

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    Well, a translation, to put "0" at "5", and then a scaling, to stretch "0 to 1" to "5 to 8".
     
  9. Oct 10, 2011 #8

    LCKurtz

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    Think about what happens if you take the "standard" equation starting at P and use PQ as the direction vector, but then replace t by (t-5)/3.
     
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