Show that all vectors in the vector field F(y,v)=(v,-y) are tangent to circles centered at the origin.

The hint I have is I can use slopes or dot product: I just need some help on how to use them.

Thanks in advance.

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- Feb 8th 2010, 04:31 PMLenGeometry of Systems
Show that all vectors in the vector field F(y,v)=(v,-y) are tangent to circles centered at the origin.

The hint I have is I can use slopes or dot product: I just need some help on how to use them.

Thanks in advance. - Feb 9th 2010, 04:21 AMshawsend
I'm gonna' give this a whirl alright. So the equation of a circle is:

$\displaystyle x^2+y^2=r^2$

differentiating:

$\displaystyle 2x+2yy'=0$

or:

$\displaystyle \frac{dy}{dx}=-\frac{x}{y}$

or parametrically:

$\displaystyle \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=-\frac{x}{y}$

and thus:

$\displaystyle \begin{aligned}

\frac{dy}{dt} &=-x \\

\frac{dx}{dt} &=y\end{aligned}

$

Also, you've got to get Mathematica or something else and just plot these to see globally what's happening. In Mathematica 7, you could use:

StreamPlot[{-x,y},{x,-3,3},{y,-3,3}]

ain't that simple or what?