Notices
Results 1 to 15 of 15

Thread: Vector Product Fromula for two >3D Vectors Found

  1. #1 Vector Product Fromula for two >3D Vectors Found 
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196


    I found the formula coupled with a method to define the cross product of two vectors in > 3D.

    It satisfies:

    Bilinearity, orthogonality, uxu = 0, anti-commutivity and a generalised length and Jacobi condition.

    I need someone's opinion - email me or post a message.


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

  2.  
     

  3. #2  
    Forum Professor river_rat's Avatar
    Join Date
    Jun 2006
    Location
    South Africa
    Posts
    1,517
    Well what is it?


    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
    Reply With Quote  
     

  4. #3 Vector prouct in >3D 
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196


    Will post the main result shortly.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  5. #4  
    Forum Freshman
    Join Date
    Aug 2007
    Location
    Cape Town, South Africa
    Posts
    8
    above 3 dimensions? im looking forward to seeing this!
    Reply With Quote  
     

  6. #5  
    Forum Professor river_rat's Avatar
    Join Date
    Jun 2006
    Location
    South Africa
    Posts
    1,517
    nothing too special about a a generalised cross product above three dimensions rich - in fact there are a host of such creatures and that is where the problem lies, the operation is no longer natural in any meaningful way.

    I'm curious if Talanum1's construction satisfies the reverse implication for parallel vectors though i.e. if u x v = 0 then u is parallel to v.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
    Reply With Quote  
     

  7. #6  
    Forum Freshman
    Join Date
    Aug 2007
    Location
    Cape Town, South Africa
    Posts
    8
    Hey, well im new to this so!

    I've never heard of a 4D cross product for example...

    I mean, what IS a 4D vector, we cant visualize it right? So how could we safely assume that a cross product is valid?

    How could a 4D vector be perpendicular to another? Unless you mean lineraly independent?
    Reply With Quote  
     

  8. #7 >3d cross product 
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196


    The generalisation of prependicularity in >3D is orthogonality. We cannot visualise it but we can test it for consistency. My definition reduces to the 3D case when the two vectors are in 3D and the unwanted fourth component is discarded.

    I can show that this definition in 4D at least is the only generalisation of the 3D one based on even and odd permutations. The determinant is defined in terms of even and odd permutations.

    I'll see if that is satisfied river_rat.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  9. #8 >3D cross product 
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196


    I have the full article for sale.

    The short version follows:

    Define operators:

    uO_i evaluates to: 1 if i odd, 0 if i even

    uE_i evaluates to: 1 if i even, 0 if i odd

    uIO_nm as assigning the numbers obtained by method 1.1 in the same

    order then evaluation to +1

    and in the opposite order then evaluation to -1

    For: [k] being the k'th unit vector, the formula is:


    uxv = [k]{ uO_k uIO_nm u_nv_m + uE_k uIO_mn u_nv_m }


    Where we sum over k (the numbr of dimensions) and over n and m as determined from the method.

    Note the reverse mn in the second uIO. The order of u_n,v_m is important: the first vector on the left must

    get the first index: in this case u gets the index n.

    Method 1.1: vector component numbers:

    1) for the numbers in dD of n,m of [1], start with the string:

    2342256227822...

    and if d is odd, end at d; if d is even write the string untill d and add a 2.

    2) complete the triangle by following the order in the following triangle:

    in 7D:

    2 3

    4 2
    3 4

    2 5
    5 3
    4 5

    6 2
    3 6
    6 4
    5 6

    2 7
    7 3
    4 7
    7 5
    6 7

    The triangle didn't come out right, but the columns can be placed next to each other to form a triangle.

    Now the numbers for [1] are 23, 42, 34, 25, 53, 45 etc.

    To get the numbers for [2]: replace 2 with 1 in the triangle of [1]
    " [3]: " 3 with 2 " [2]

    Continue untill the numbers for all of [k] is determined.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  10. #9  
    Forum Freshman
    Join Date
    Aug 2007
    Location
    Cape Town, South Africa
    Posts
    8
    ok i have no idea what that means

    so is this really valid?
    Reply With Quote  
     

  11. #10  
    Forum Professor river_rat's Avatar
    Join Date
    Jun 2006
    Location
    South Africa
    Posts
    1,517
    question - why is the article for sale?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
    Reply With Quote  
     

  12. #11  
    Forum Freshman
    Join Date
    Aug 2007
    Location
    Cape Town, South Africa
    Posts
    8
    yeah thats what i wondered?
    Reply With Quote  
     

  13. #12  
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    :?

    It is for sale because I don't work at a university.

    I can show that uxv = 0 implies u parallel to v. Will post this shortly.

    Be more specific about what you don't understand please.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  14. #13  
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    :?

    [1] is the 1'st unit vector.

    [1] uO_1 uIO_nm v_nu_m = [1] uIO_nm v_nu_m (1)

    since uE_1 = 0

    [2] uE_2 uIO_nm v_nu_m = [2] uIO_nm v_nu_m (2)

    since uO_2 = 0

    Now the number in [] tells you what set of numbers to sum v_nu_m over.

    [1]' s numbers are determined by the method. In 4D this is

    23, 42, 34.

    Expand (1):

    [1] uIO_nm v_nu_m + [1] uIO_nm v_nu_m + [1] uIO_nm v_nu_m

    + [1] uIO_nm v_nu_m + [1] uIO_nm v_nu_m + [1] uIO_nm v_nu_m

    Now plug in the numbers in order, one pair into each n,m and change uIO_mn into +1 if assigning them in the same order and into -1 if in the opposite order.

    Do the same expansion for (2) and plug in the numbers:

    31, 14, 43

    in the same fasion. The numbers for even unit vectors are copied from the method in reverse order:

    13 > 31, 41 > 14, 34 > 43

    before plugging them in.

    If you have the method for one odd and one even unit vector and you have the method for getting the numbers then you have the method for any unit vector.

    I proved the properties as in previous messages in my article.

    I can post a vew of the proprty proofs. Please indicate wich one is the most important.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  15. #14  
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    8)

    The equivalent formulation (easier to use is):

    uxv = [k]{ uO_k uIO_nm u_nv_m - uE_k uIO_nm u_nv_m }

    and now you don't have to reverse the numbers of the even unit vectors before inserting them.

    river_rat: The proof is easy: Form theorem quoted:

    if vxu = 0 then u and v are liniarly dependent. Therefore v can be written as au, a>0, a real.

    Now au is parrallel to u and any vector v parallel to au is just au with the n-tuple like [b,b,b,b..b] added to both the start and endpoint of au.

    We have uxu = 0, auxu = 0 therefore au is parrallel to u,

    now replace au with v in the last sentence.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  16. #15  
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196


    The same formula can actually be stated in terms of a determinant (of sorts) that reduces to a sum of 3 by 3 determinants.
    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
  •