Higher-dimensional spaces curiosity
1 Answer
I guess by intersection of 3D space in $R^4$ you mean intersection of two ahyperplanes:
\[a_1x+b_1y+c_1z+d_1w=e_1 \text{and} a_2x+b_2y+c_2z+d_2w=e_2.\]
It is possible for these two hyperplanes to have (I) no intersection or (II) coincide. However, if cases (I) and (II) does not happen, then using basic facts from linear algebra (you may wire $w$ in terms of the other three variables in one equation and replace in the other) one can conclude that the intersection will be a two-dimensional plane.
You conjecture is true in any dimension $n$, and can be proved with a very similar argument.

133
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- 1 Answer
- 274 views
- Pro Bono
Related Questions
- Geometric Representation Problem
- Land area calculation/verification
- A trigonometry question
- Geometric Representation Question
- Find the area of the shaded region
- Can a plane reflection in 3-space be written as the product of 3 line reflections?
- Can enough pizza dough be made to cover the surface of the earth?
- Get area of rotated polygon knowing all coordinates and angle.