# Higher-dimensional spaces curiosity

## 1 Answer

\[a_1x+b_1y+c_1z+d_1w=e_1 \text{and} a_2x+b_2y+c_2z+d_2w=e_2.\]

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.

Poincare

123

Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

- 1 Answer
- 186 views
- Pro Bono

### Related Questions

- Pulley System
- Can enough pizza dough be made to cover the surface of the earth?
- Probability that the distance between two points on the sides of a square is larger than the length of the sides
- Get area of rotated polygon knowing all coordinates and angle.
- Land area calculation/verification
- Geometry Problem about Hole Placement on PVC Pipe
- Find the area of the shaded region
- Volume of a sphere.