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Math Help - Vector Calculus (Position Vector)

  1. #1
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    Vector Calculus (Position Vector)

    The position vector of a point is given by \vec{r}(t) = (e^tcost, e^tsint). Prove that:

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

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


    Answer: b) pi/2

    ____________________________________________

    I tried to derive the position vector two times but it isn't working.
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  2. #2
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    Re: Vector Calculus (Position Vector)

    Quote Originally Posted by PedroMinsk View Post
    The position vector of a point is given by \vec{r}(t) = (e^tcost, e^tsint). Prove that:
    a) \vec{a} = 2\vec{v} - 2\vec{r}
    b) the angle between the position vector \vec{r} and the acceleration vector \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 e^t\sin(t).
    That will help you find \vec{a} the second derivative or \vec{r}
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  3. #3
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    Re: Vector Calculus (Position Vector)

    For the second question you can calculate the scalar product of the vectors a(x_1,y_1),r(x_2,y_2) like:
    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).
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    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?
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  5. #5
    MHF Contributor Siron's Avatar
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    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 \arccos(0)=\frac{\pi}{2}\right)( +2k\pi)
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    Re: Vector Calculus (Position Vector)

    Quote Originally Posted by PedroMinsk View Post
    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.
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  7. #7
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    Re: Vector Calculus (Position Vector)

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