Thread: 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.

Have you read the question correctly? What's the reference? The statement should be that is part of a line through the origin; it doesn't have to be the entire line.

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

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

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 you can get as close to as you like.

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 you can get as close to 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.

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.