3d surface in 4d space

  1. 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. jcsd
  3. Mark44

    Staff: Mentor

    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.
    In higher dimensions, including R4, we call them hyperplanes. And again, a single vector isn't enough.
     
  4. HallsofIvy

    HallsofIvy 40,367
    Staff Emeritus
    Science Advisor

    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.
     
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