Register to reply

3d surface in 4d space

by dEdt
Tags: space, surface
Share this thread:
dEdt
#1
Jan17-14, 04:55 PM
P: 282
I hope this is the right forum...

In 3d space, a 2d plane can be specified by it's normal vector. In 4d space, is there a 3d plane, and will these planes be specifiable by a single vector?
Phys.Org News Partner Mathematics news on Phys.org
Professor quantifies how 'one thing leads to another'
Team announces construction of a formal computer-verified proof of the Kepler conjecture
Iranian is first woman to win 'Nobel Prize of maths' (Update)
Mark44
#2
Jan17-14, 07:10 PM
Mentor
P: 21,272
Quote Quote by dEdt View Post
I hope this is the right forum...

In 3d space, a 2d plane can be specified by it's normal vector.
No, that's not enough information. You can specify a plane in R3 by its normal vector and a point on the plane. Without that point what you get is a family of parallel planes.
Quote Quote by dEdt View Post
In 4d space, is there a 3d plane, and will these planes be specifiable by a single vector?
In higher dimensions, including R4, we call them hyperplanes. And again, a single vector isn't enough.
HallsofIvy
#3
Jan17-14, 10:57 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,510
In general, we can specify a n-1 dimensional hyperplane in a space of n dimensions with a "normal vector" and a point in the hyperplane.

In four dimensions, every point can be written as [itex](x_1, x_2, x_3, x_4)[/itex] and a four dimensional vector of the form [itex]a\vec{ix}+ b\vec{j}+ c\vec{k}+ d\vec{l}[/itex]. If the origin, (0, 0, 0, 0) is in the hyperplane, then we can write [itex]x_1\vec{ix}+ x_2\vec{j}+ x_3\vec{k}+ x_4\vec{l}[/itex] and so the dot product is [tex]ax_1+ bx_2+ cx_3+ dx_4= 0[/tex] giving an equation for that hyper plane. But, again, that assumes the hyperplane contains the point (0, 0, 0). Another plane, perpendicular to the same vector, but not containing (0, 0, 0), cannot be written that way.


Register to reply

Related Discussions
Fun with Surface Tension in Space Chemistry 1
3d phase space-surface Differential Equations 2
What is a realization of this surface in Euclidean space? Differential Geometry 14
How much is space bended at the surface of the sun Special & General Relativity 1
Free-fall from space to earth surface Advanced Physics Homework 14