# vector function question

• Jun 24th 2011, 04:23 PM
kkoutsothodoros
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.
• Jun 24th 2011, 04:41 PM
ojones
Re: vector function question
Have you read the question correctly? What's the reference? The statement should be that $\displaystyle \mathbf{r}(t)$ is part of a line through the origin; it doesn't have to be the entire line.
• Jun 24th 2011, 05:04 PM
kkoutsothodoros
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.
• Jun 24th 2011, 05:06 PM
kkoutsothodoros
Re: vector function question
please someone just tell me the question has a typo because this is driving me a little nutty.
• Jun 24th 2011, 05:10 PM
ojones
Re: vector function question
Quote:

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 $\displaystyle t\rightarrow -\infty$ you can get as close to $\displaystyle (0,0)$ as you like.
• Jun 24th 2011, 05:56 PM
kkoutsothodoros
Re: vector function question
Quote:

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 $\displaystyle t\rightarrow -\infty$ you can get as close to $\displaystyle (0,0)$ 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.
• Jun 24th 2011, 06:11 PM
ojones
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.