Notices
Results 1 to 3 of 3

Thread: Visualising vectors in 4D and beyond!

  1. #1 Visualising vectors in 4D and beyond! 
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    I found a way to visualise 4D vectors! - Very simple.

    Just take a 4D vector u and draw it in 3D as if the 4'th component doesn't matter,
    (i.e. ignore the 4'th component). Then draw its "ghost" or image u' in the same axis
    system with just the 4'th and 1'st component "interchanged" via the cross product
    as follows:

    for two 4D vectors v, w let the result of vxw be u, then

    u' = [1R4](vxw)_1[1] - (vxw)_2[2] - (vxw)_3[3] + [4R1](vxw)_4[4]

    where the [1R4] means index 1 replaces index 4 in the operand in ( ) after taking
    the m'th component for each _m, and [n] is the n'th unit vector.

    This formula comes from:

    u' =

    [1] [2] [3] [4]
    v_4 v_2 v_3 v_1
    w_4 w_2 w_3 w_1


    \textibf {u'} =
    \left|
    \begin {array}
    {r r r r}
    i&j&k&l\\
    v_4&v_2&v_3&v_1\\
    w_4&w_2&w_3&w_1\\
    \end {array}\right|

    and use my definition for the 4D cross product (posted here in December 2007).

    There is a way to get some v and w from u (will be posted soon).

    One can also exchange 4'th and 3'rd, or 4'th and 2'nd but since we give the axises
    consecutive labels the usual convention should be to exchange the 4'th and 1'st.

    One can now re-look at the light cone of 3D + 1D of relativity for the case where
    the particle (or everything) is not restricted to move in a 2D plane.

    For larger dimensions (n) one would have one restricted and n-3 ghost vectors.

    Edited 25072008.

    Thanks Jane.


    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  2.  
     

  3. #2 Re: Visualising vectors in 4D and beyond! 
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    Quote Originally Posted by talanum1
    (the bold button does not work neither does Tex)
    Did you accidentally tick the “Disable BBCode in this post” box below the text area where you type your message? If so, leave the box blank.

    BBCode and TeX are both working fine for me.


    Reply With Quote  
     

  4. #3  
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    Actually the following is a better way since it does not discard anything:

    plot u and:

    u' = u_4[1] - u_2[2] - u_3[3] + u_1[4]

    where [n] is the n'th unit vector. The sign reversals are required so that the two
    does not overlap if u_4 is zero.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •