Proving Magnutude Of Vector Valued Function Is Constant

[Lancelot59]

Proving Magnitude Of Vector Valued Function Is Constant

I have a rather neat problem. I need to prove that the magnitude of this function:

F_{2}(x,y)=\frac{<-x,y>}{x^{2}+y^{2}}

is constant along circles centred about the origin. Now while proving that the magnitude is inversely proportional I had to get the magnitude, and it wound up being:

\sqrt{\frac{1}{(x^{2}+y^{2})}}

Which looks like the basic equation for a circle to me. That particular function will give elliptical level curves. I'm not sure how to go about this. Can I just say that because it's the inverse of a the form of an ellipse that if the level curve x^2+y^2 is equal to a constant c^2 then the magnitude will also be a constant because it has the same form?

 

[Char. Limit]

that's not what I get for the magnitude. Can you show me how you got that answer?

 

[Lancelot59]

This is how I did it:

First I multiplied in the scalar x^2+y^2.
=<\frac{-x}{x^{2}+y^{2}},\frac{y}{x^{2}+y^{2}}>

Then put it in the usual form:
=\sqrt{(\frac{-x}{x^{2}+y^{2}})^{2}+(\frac{y}{x^{2}+y^{2}})^{2}}

They have a common denominator, so I added them:
=\sqrt{\frac{x^{2}+y^{2}}{(x^{2}+y^{2})^{2}}

Which then simplifies down to:
=\sqrt{\frac{1}{x^{2}+y^{2}}

 

[Char. Limit]

Whoops, my bad.

Well, your magnitude is equal to \frac{1}{r}, isn't it? Seems pretty obvious to me.

 

[Lancelot59]

Whoops, my bad.

Well, your magnitude is equal to \frac{1}{r}, isn't it? Seems pretty obvious to me.

I see what you did there. Thanks!