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Thread: Can you help me build a simple formula?

  1. #1
    Newbie Flightline's Avatar
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    Can you help me build a simple formula?

    I'm not in school, and I'm not looking for any answer to any homework assignment. I have something I'm doing on my own, and I wanted to see if anyone comes up with the same formula, or close to it, as I did for the following.

    Take a graph: draw a line so that $\displaystyle x=y$, which we know is a $\displaystyle 45^o$angle and a $\displaystyle 135^o$ angle from $\displaystyle (0,0)$ heading in a positive and negative direction on to infinity.

    Now, let's say that the line goes to $\displaystyle (1,1)$ then $\displaystyle (2,2)$ then $\displaystyle (3,3)$ etc. What formula would also extend the line to $\displaystyle (-1,-1) $ $\displaystyle (-2,-2) $ $\displaystyle (-3,-3)$ ...ect., as the line extends positively?

    In other words, you start from $\displaystyle (0,0)$ and the vectors extend positively, and negatively in proportion to each other. When the vector goes from $\displaystyle (0,0)$ to $\displaystyle (1,1)$ to $\displaystyle (2,2)$ it then extends to at the same time to $\displaystyle (-1,-1)$, $\displaystyle (-2,-2)$, etc.

    I understand people have their own work and may not want to be sidetracked with this, but I do appreciate anything anyone has to suggest. My guess is this is a well-known formula, and I'm just don't know well enough to know it. Nevertheless, I want to get this simple formula down before proceeding on with my work.

    Thank you very much in advance for anything.

    Ed
    Last edited by Flightline; Jul 27th 2010 at 10:10 PM.
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  2. #2
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    Hello !

    The same formula is suitable $\displaystyle \ \ y=x \ \ $
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  3. #3
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    $\displaystyle y=x$ goes through all the points mentioned above
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  4. #4
    Newbie Flightline's Avatar
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    Thank you, I thought about that, that simply X=Y would do, but that produces an infinite line in both directions. I was looking for a formula, or call it a model, if you will, that indicats that as the line goes to 1,1 it also goes to -1,-1. I have a formula I'd like to show here, but I'm not sure how, just yet. But I will try using TEX, and do a bunch of edits and see what happens. Here it goes:

    $\displaystyle
    \left(f_(_x_)= x | x=\sum\limits_{n=0}^{\infty}n+1\right)\Longrightar row \left(f_(_x_)= x | x=\sum\limits_{n=(-1)}^{\infty}n-1\right)$
    Last edited by Flightline; Jul 27th 2010 at 11:12 PM.
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  5. #5
    Newbie Flightline's Avatar
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    OK, above is the formula I have come up with.
    Jeez...talk about a crash course in LaTex
    In your opinion, what is wrong with this formula? does it do what I am wanting it to do?

    I don't know. I think I don't need the summation symbols. I'm not summing an infinite series for (x,y). I'm just trying to say that if (x,y) is such and such a coordinate, then the line moves from zero as well to the negative of that coordinate.

    I should have waited to post this until I worked with it. Oh well, it was good LaTex practice.

    Again, I realize this takes up your time, so any comments I appreciate.
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  6. #6
    Senior Member yeKciM's Avatar
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    $\displaystyle \displaystyle (\forall x,y \in \mathbb{R} ) (y=x) $

    it's just a line, and for every x witch goes from $\displaystyle \displaystyle -\infty$ to $\displaystyle \displaystyle+\infty$ u'll have $\displaystyle \displaystyle y=x$ ....
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  7. #7
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    Quote Originally Posted by Flightline View Post
    Thank you, I thought about that, that simply X=Y would do, but that produces an infinite line in both directions. I was looking for a formula, or call it a model, if you will, that indicats that as the line goes to 1,1 it also goes to -1,-1. I
    I'm not certain what you mean by "as the line goes to 1,1 it also goes to -1,-1". A line doesn't "go" to one point or another, it just is there. Perhaps what you mean is a set of functions: $\displaystyle f_X(x)= x$ for $\displaystyle -X\le x\le X$. Or a function of two variables: $\displaystyle f(x,t)= x$ for $\displaystyle -t\le x\le t$.
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  8. #8
    Newbie Flightline's Avatar
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    Thanks for the input, and now I wish I had waited before posting about this until I worked it out better. Nevertheless, I know what $\displaystyle x = y $ means, or at least what it looks like. It's a line that runs infinitely in both directions from $\displaystyle (0,0)$, i.e., $\displaystyle \{...-3,-2,-1,0,1,2,3...\}$

    What I want to show is that there is a force starting at point $\displaystyle (0,0)$. If the force increases in magnitude in one direction, there is a corresponding increase in the other direction. Thus if the increase in magnatude is represented by a line going from $\displaystyle (0,0)$ to $\displaystyle (1,1)$ in the first quadrant of the graph, the line also increases in the third quadrant of the graph from $\displaystyle (0,0)$ to $\displaystyle (-1,-1)$.

    $\displaystyle x=y$ is not the formula that illustrates this, or at least it doesn't seem that way to me. Because as has been pointed out, $\displaystyle x=y $ or other forms of saying the same thing such as $\displaystyle \displaystyle (\forall x,y \in \mathbb{R} ) (y=x)$ merely stand for a line that goes from infinity to infinity along that vector.
    Last edited by Flightline; Jul 28th 2010 at 08:15 AM.
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  9. #9
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    edit wrong answer
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  10. #10
    Newbie Flightline's Avatar
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    After some more thought, I came up with this. What do you think?

    $\displaystyle
    \left(f_(_x_)= x | x=n+0)\Longrightarrow \left\left(f_(_x_)= x | x=0-n)$

    If $\displaystyle y=x$ such that $\displaystyle x = \{1,2,3,...\}$; then $\displaystyle y=x$ such that $\displaystyle x=\{-1,-2,-3...\}$
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  11. #11
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    this is all very confusing. Why are you trying to find an equation to draw 2 vectors at once?

    If your objection to y=x is that it is unbounded, then you can simply add a boundary
    $\displaystyle y=x : -c < x < c $

    Alternatively, graph the vectors seperately:
    Vector 1: (c,c)
    Vector 2: -(c,c)
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  12. #12
    Junior Member bondesan's Avatar
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    Flightline,

    Maybe you should see this blog post - the equation of a line segment. I think this is what you're looking for.

    The Equation of a Line Segment

    I hope it helps.
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  13. #13
    Newbie Flightline's Avatar
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    Quote Originally Posted by SpringFan25 View Post
    this is all very confusing. Why are you trying to find an equation to draw 2 vectors at once?

    If your objection to y=x is that it is unbounded, then you can simply add a boundary
    $\displaystyle y=x : -c < x < c $

    Alternatively, graph the vectors seperately:
    Vector 1: (c,c)
    Vector 2: -(c,c)
    Thanks, but I don't think that will work for what I'm trying to model. I'm trying to model the idea that as a vector progresses one way, it causes an equal and opposite progression the other way. As to why I'm doing it, well, I'm not sure. I think it represents a concept in a larger concept I'm working on, but since it's probably all a delusion of grandeur anyway, I'd rather not embarrass myself by talking about it.
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  14. #14
    Newbie Flightline's Avatar
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    Quote Originally Posted by bondesan View Post
    Flightline,

    Maybe you should see this blog post - the equation of a line segment. I think this is what you're looking for.

    The Equation of a Line Segment

    I hope it helps.
    I actually think that article will help me a lot in my understanding of what I'm trying to do. Thank you very much for leading me to it. I'm going to give it a good read in the bathtub--the place where all great ideas are born!
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  15. #15
    Junior Member bondesan's Avatar
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    Hahaha, good one. You are welcome.
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