# Vector Calculus (Position Vector)

• Aug 23rd 2011, 12:18 PM
PedroMinsk
Vector Calculus (Position Vector)
The position vector of a point is given by $\displaystyle \vec{r}(t) = (e^tcost, e^tsint)$. Prove that:

a) $\displaystyle \vec{a} = 2\vec{v} - 2\vec{r}$

b) the angle between the position vector $\displaystyle \vec{r}$ and the acceleration vector $\displaystyle \vec{a}$ is constant. Calculate this angle.

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I tried to derive the position vector two times but it isn't working.
• Aug 23rd 2011, 01:06 PM
Plato
Re: Vector Calculus (Position Vector)
Quote:

Originally Posted by PedroMinsk
The position vector of a point is given by $\displaystyle \vec{r}(t) = (e^tcost, e^tsint)$. Prove that:
a) $\displaystyle \vec{a} = 2\vec{v} - 2\vec{r}$
b) the angle between the position vector $\displaystyle \vec{r}$ and the acceleration vector $\displaystyle \vec{a}$ is constant. Calculate this angle.

This is a tedious problem.
Luckily there are web resources
BE SURE to click "show steps"
You can change the question to $\displaystyle e^t\sin(t)$.
That will help you find $\displaystyle \vec{a}$ the second derivative or $\displaystyle \vec{r}$
• Aug 23rd 2011, 01:30 PM
Siron
Re: Vector Calculus (Position Vector)
For the second question you can calculate the scalar product of the vectors $\displaystyle a(x_1,y_1),r(x_2,y_2)$ like:
$\displaystyle a.r=x_1x_2+y_1y_2$
The scalar produt will tell you more about the angle between the vectors, but first you've to determine the coordinates of the vector r(t) (see Plato's post).
• Aug 23rd 2011, 01:34 PM
PedroMinsk
Re: Vector Calculus (Position Vector)
Thanks. I got the idea. I only knew the Wolfram integrator, this one will help me a lot.

Can I calculate the angle using dot product? Or there is another way?
• Aug 23rd 2011, 01:37 PM
Siron
Re: Vector Calculus (Position Vector)
The dot (or scalar) product is very useful here, you'll come to the conclusion if you calculate the dot product that it will be 0 and $\displaystyle \arccos(0)=\frac{\pi}{2}\right)$($\displaystyle +2k\pi$)
• Aug 23rd 2011, 01:39 PM
Plato
Re: Vector Calculus (Position Vector)
Quote:

Originally Posted by PedroMinsk
Thanks. I got the idea. I only knew the Wolfram integrator, this one will help me a lot.
Can I calculate the angle using dot product? Or there is another way?

Yes the dot product, see reply #3.
You should explore what all wolframalpha will do.
Look at this.
• Aug 23rd 2011, 01:43 PM
PedroMinsk
Re: Vector Calculus (Position Vector)
I got it. Thanks. I will see the link.