# Proof involving central acceleration and vector products

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1. Sep 8, 2016

### Sho Kano

1. The problem statement, all variables and given/known data
Suppose $r:R\rightarrow { V }_{ 3 }$ is a twice-differentiable curve with central acceleration, that is, $\ddot { r }$ is parallel with $r$.
a. Prove $N=r\times \dot { r }$ is constant
b. Assuming $N\neq 0$, prove that $r$ lies in the plane through the origin with normal $N$.

2. Relevant equations

3. The attempt at a solution
a. $\frac { d }{ dt } N=\frac { d }{ dt } r\times \dot { r } =r\times \ddot { r } +\dot { r } \times \dot { r } =\overrightarrow { 0 }$ because $r$ is parallel with $\ddot { r }$

b. $\dot { r }$ is in the same plane as $r$, then the equation of the plane through the origin is $\left< x,y,z \right> \cdot r\times \dot { r } =0$. If $r=\left< x,y,z \right>$, then $r\cdot r\times \dot { r } =r\times r\cdot \dot { r } =0$ which checks out

I'm really not sure if I'm right

2. Sep 8, 2016

### LCKurtz

It looks correct.

3. Sep 9, 2016

### Sho Kano

Awesome, I can turn in my homework at ease now. Thanks.