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A Challenging Idea

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I have this idea that is very interesting, but I don't have any time to pursue it at the moment. Enjoy racking your brain!

Say I have a vector in 3-D space defined as r1 = a1i + b1j + c1k, where i,j and k are the standard basis in 3D vector space and a,b,c are continuous, differentiable functions with respect to some parameter t. (I'm trying to write R sub 1, a sub 1 etc...)

So now let pr be the vector pr1 = pa1i + pb1j + pc1k such that each component of the vector has a probability p of appearing and a probability 1-p of having a value of 0.

Now, imagine a set of vectors r1, r2 ...rn such that r1 + r2 + r3 + ... rn = 0 and no vector intersects any other - they form a directed planar graph.

Questions:

1. for p = 0.5, what does this look like geometrically?

2. for p = 0.5, what does its 1st derivative planar graph look like?

3. what happens when p is a function of t?

4. what about in 4D space? n-tuple space?

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That's pretty crazy. do you mean that r1, r2 etc are functions of x, y, and z? we're currently studying vectors in HL maths, so I might have a think about your idea, but I've a lot of work to do at the moment.

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I'm rather confused about the definition of the problem, but how can the vectors form a directed planar graph? They aren't points nor lines, rather the definition seems to just give it a direction. I can' see how they won't intersect either since they all lie on the same plane.

Also, where did you get this problem from? It seems rather ambiguous and overly general to properly imagine, at least with my current knowledge of vectors. Does t stay constant? Are the functions a1 a2 ... an related in some form, or are they just bound by the condition of making a planar graph?

Sounds like an interesting problem, but I'm struggling to even get an idea of what it's really about.

That's pretty crazy. do you mean that r1, r2 etc are functions of x, y, and z? we're currently studying vectors in HL maths, so I might have a think about your idea, but I've a lot of work to do at the moment.

Not exactly, Rn is made up of an, bn and cn, which are functions with a parameter t.

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I'm rather confused about the definition of the problem, but how can the vectors form a directed planar graph? They aren't points nor lines, rather the definition seems to just give it a direction. I can' see how they won't intersect either since they all lie on the same plane.

Also, where did you get this problem from? It seems rather ambiguous and overly general to properly imagine, at least with my current knowledge of vectors. Does t stay constant? Are the functions a1 a2 ... an related in some form, or are they just bound by the condition of making a planar graph?

Sounds like an interesting problem, but I'm struggling to even get an idea of what it's really about.

That's pretty crazy. do you mean that r1, r2 etc are functions of x, y, and z? we're currently studying vectors in HL maths, so I might have a think about your idea, but I've a lot of work to do at the moment.

Not exactly, Rn is made up of an, bn and cn, which are functions with a parameter t.

sorry I meant to ask if an, bn etc. are functions of x, y and z? or are functions of t?

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It's not a textbook problem or contest question. I was just developing this idea a bit and decided to throw it out for anyone else to see. I should specify something here. When I say a directed planar graph, I was thinking about a project onto a 2D surface from whatever dimension the original comes from. That aside, I do not know the dependencies of the coefficients of the vectors (do we really want to put them into a coordinate system? technically, vectors can exist without a fixed coordinate system).

Of course, the direction of the perspective is important as well. There will be certain ways to take the projection of the said vectors in such a way that the projection is a line, or a shape of some sort. I'm not too sure how to approach this problem. Keep thinking!

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