# Need Help with Newton's equation and circular motion

## 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 independant

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 independant

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!