Register to reply 
3d surface in 4d space 
Share this thread: 
#1
Jan1714, 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? 


#2
Jan1714, 07:10 PM

Mentor
P: 21,272




#3
Jan1714, 10:57 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,510

In general, we can specify a n1 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 spacesurface  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  
Freefall from space to earth surface  Advanced Physics Homework  14 