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Math Help - vector function question

  1. #1
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    vector function question

    does it necessarily follow from dr/dt = ar(t) that r(t) must trace out a straight line through the origin? r(t) is a vector function and 'a' is a scalar constant.

    i have a counterexample: r(t)={2exp(at), 3exp(at)}. r(t) cannot be {0, 0} because of properties of exp(at).

    this is a qustion in a calculus book. so i am assuming i am missing something by finding a counterexample.
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  2. #2
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    Re: vector function question

    Have you read the question correctly? What's the reference? The statement should be that \mathbf{r}(t) is part of a line through the origin; it doesn't have to be the entire line.
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  3. #3
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    Re: vector function question

    r(t) is the position vector. the question asks to prove that the particle's trajectory is a straight line through origin.
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  4. #4
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    Re: vector function question

    please someone just tell me the question has a typo because this is driving me a little nutty.
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  5. #5
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    Re: vector function question

    Quote Originally Posted by kkoutsothodoros View Post
    please someone just tell me the question has a typo because this is driving me a little nutty.
    I have to see the question before I can decide if there's typo. In any event, what I think the question means is that the trajectory is a line through the origin, but doesn't include the origin. If you let t\rightarrow -\infty you can get as close to (0,0) as you like.
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  6. #6
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    Re: vector function question

    Quote Originally Posted by ojones View Post
    I have to see the question before I can decide if there's typo. In any event, what I think the question means is that the trajectory is a line through the origin, but doesn't include the origin. If you let t\rightarrow -\infty you can get as close to (0,0) as you like.
    exact question as written in Richard A. Silverman's Modern Calculus and Analytic Geometry:

    suppose the radius vector r (bold vector notation) = r(t) of a moving particle satisfies the differential equation dr/dt (bold vector r) = ar(t) where a is a scalart constant. Prove the particle's trajectory is a straight line through the origin.

    I don't see how I've misinterpreted. But Silverman's books are good so I tend to trust.
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  7. #7
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    Re: vector function question

    There seems to be a small ommission in Silverman's question. The solution you found is correct and you're right in that it doesn't pass through the origin. The trajectory is a line through the origin but doesn't include the origin.
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