Proving Magnutude Of Vector Valued Function Is Constant

In summary, the conversation discusses the problem of proving that the magnitude of a vector valued function is constant along circles centered about the origin. The function is F_{2}(x,y)=\frac{<-x,y>}{x^{2}+y^{2}} and the magnitude is shown to be equal to \sqrt{\frac{1}{x^{2}+y^{2}}}. The conversation also mentions using the common denominator and simplifying to \frac{1}{r}, which proves the magnitude is indeed constant.
  • #1
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Proving Magnitude Of Vector Valued Function Is Constant

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

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

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:

[tex]\sqrt{\frac{1}{(x^{2}+y^{2})}}[/tex]

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?
 
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  • #2
that's not what I get for the magnitude. Can you show me how you got that answer?
 
  • #3
This is how I did it:

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

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

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

Which then simplifies down to:
[tex]=\sqrt{\frac{1}{x^{2}+y^{2}}[/tex]
 
  • #4
Whoops, my bad.

Well, your magnitude is equal to [tex]\frac{1}{r}[/tex], isn't it? Seems pretty obvious to me.
 
  • #5
Char. Limit said:
Whoops, my bad.

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

I see what you did there. Thanks!
 

What is a vector valued function?

A vector valued function is a mathematical function that maps a set of inputs to a set of vectors as outputs. It is often represented by a vector with multiple components, each of which is a function of the same input variable.

What is magnitude of a vector?

The magnitude of a vector is a measure of its length or size. It is calculated by taking the square root of the sum of the squares of its components. In other words, it represents the distance from the origin to the tip of the vector in a coordinate system.

How do you prove that the magnitude of a vector valued function is constant?

In order to prove that the magnitude of a vector valued function is constant, you need to show that the magnitude remains the same for all values of the input variable. This can be done by taking the derivative of the function and showing that it is equal to zero, indicating that the magnitude is not changing.

What is the significance of proving that the magnitude of a vector valued function is constant?

Proving that the magnitude of a vector valued function is constant can have several implications. It can show that the vector is always pointing in the same direction, or that its length does not change over time. This can be useful in various fields such as physics, engineering, and computer graphics.

What are some real-life applications of proving the magnitude of a vector valued function is constant?

One example of a real-life application of this concept is in motion planning for autonomous vehicles. By proving that the magnitude of the velocity vector is constant, it can be ensured that the vehicle will maintain a constant speed and direction. This can also be useful in analyzing the motion of objects in physics, such as projectiles or planets in orbit.

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