- #1
JulienB
- 408
- 12
Homework Statement
Hi! I'm not sure the title I gave properly reflects my problem, but I did not know how to describe it in another way:
(roughly translated from German)
"In three dimensions, we can divide the space in two parts through a plane. In four dimensions, that does not work anymore. Given the plane in ℝ4:
E = { (x1, x2, 0, 0) | x1, x2 ∈ ℝ}
Show that two arbitrary points in ℝ4 not belonging to E can be joined by a maximum of two lines without intersecting with E."
Homework Equations
What gives me a headache is that we have not even started to talk about higher dimensions than 3 yet. Our teacher encourages us through the homework to make some research and learn on our own, but I could not find any relevant information about how to deal with such a problem.
However, we have mentioned scalar multiplication in ℝn and the fact that two coordinates of the plane are 0 makes me think that this problem might be easier than it seems.
The Attempt at a Solution
I have no idea where to start :( So far I only tried to formulate all I know in an algebraic way to avoid trying to "imagine" what four spatial dimensions might imply.
So I first defined two points:
P1, P2 ∈ ℝ4 ∧ P1, P2 ∉ E
For two lines to exist (and forget about the trivial case where one line would be enough), there must be a third point:
P3 ∈ ℝ4, ∉ E
And for ∀ Pi ∈ ℝ4, the lines A (P1, P2) and B (P2, P3) ∉ E
Unfortunately that didn't lead me anywhere... Any advice or suggestion of what to search for would be very much appreciated.Thank you in advance for your answers.J.