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.

Answer: b) pi/2

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I tried to derive the position vector two times but it isn't working.

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 $\displaystyle \vec{r}$

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).

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?

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 $\displaystyle \arccos(0)=\frac{\pi}{2}\right)$($\displaystyle +2k\pi$)

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.

Re: Vector Calculus (Position Vector)

I got it. Thanks. I will see the link.