1. ## 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.

2. ## 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.

3. ## 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.

4. ## Re: vector function question

please someone just tell me the question has a typo because this is driving me a little nutty.

5. ## Re: vector function question

Originally Posted by kkoutsothodoros
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.

6. ## Re: vector function question

Originally Posted by ojones
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.

7. ## 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.