Need Help with Newton's equation and circular motion

[JPTM]

Homework Statement

1. Proof that the vectors $\vec{t}(t) = cos(ω(t)), sin(ω(t))$ and $\vec{n}(t) = -sin(ω(t)), cos(ω(t))$ are linearly independent

2. Proof that $ω = ω\in\mathbb R$

3. Can all angular velocities indicate a possible circular orbit? If so, proof it, if not which angular velocities do?

-ω is the angular velocity
-The movement starts at (r,0)
-We don't use r = 0

Homework Equations

We have 3 given equations:

$$F(t) = μ\vec{r''}(t) = - \dfrac{k}{(r(t))^3}\vec{r}(t)$$

Which can be rewritten as:

$$\vec{r''}(t) = \dfrac{f(r(t))}{μr(t)}\vec{r}(t)$$

Also the equation of our circular motion is:

$$\vec{r}(t) = r(cos(ω(t)), sin(ω(t)))$$

The Attempt at a Solution

1. Can simply be proven by calculating the dot product of t(t) and n(t) which = 0 which means that they have to be linearly independent

2. This is the one I'm stuck at, I've never had any exercise or read something about how you could proof that something is a real number

3. I think I need 2. for this one

Help would be grately appreciated!

 

[tia89]

With order...

Pay attention that in point 1) you are simply proving, showing that the dot product is zero, that the vectors are perpendicular. You have to prove that they are linearly independent though, which I don't think in general is the same thing. Use the definition of linearly independent vectors and show that indeed the combination $a\vec{t}+b\vec{n}=0$ if and only if $a$ and $b$ are both zero (it is easily done). If it happens then indeed they will be linearly independent.

As for point 2), I do not understand what is the meaning of such a question. In general the argument of a sine or cosine is real, so you should have some condition to satisfy for this not to happen. And I also do not understand the meaning of the third question as well... could you give some more information about this?? Also perhaps specifying what the equations you have are??

 

[voko]

Orthogonality always implies linear independence, so I think the proof of #1 is OK.

The other questions are indeed puzzling.