On { N+1 Symmetric Lines } in a { N-Dimension Vector Space }

by Frank C. Fung ( 1st published in March, 2016. )

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Section III - { 4 Symmetric Lines } in a { 3-Dimension Space }

Summary for the Section :

• In this Section , we shall identify , via the [ cube ] :
  • { 4 Symmetric Lines } in a { 3-Dimension Space } .

  We shall also list-out an alternate procedure for :

  • creating the { 4 Symmetric Lines } in a { 3-Dimension Space } ;

    for the purpose of comparison with the procedure we shall later list-out :

    • in Section IV for { 5-Symmetric Lines in a 4-Dimension Space } ,

    • in Section V for { 6-Symmetric Lines in a 5-Dimension Space } .

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A quick look at the Cube :

• Let us now take a quick look at the Cube :

  

  We see here that :

  • the 4 lines joining opposite corners of the Cube are in fact symmetric

    and

    do qualify as a set of { 4 Symmetric Lines } in a { 3-Dimension Space } .

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Creating the { 4 Symmetric Lines } via an alternate procedure :

• Let us now list-out an alternate procedure for :
  • creating { 4 Symmetric Lines } in a { 3-Dimension Vector Space } .

  And here are the steps to do so :

  STEP ONE :

  • We first create a { 4-Dimension Vector Space } defined by the 4 { unit vectors } :
    • { 1, 0, 0, 0 }
    • { 0, 1, 0, 0 }
    • { 0, 0, 1, 0 }
    • { 0, 0, 0, 1 }

  STEP TWO :

  • We then identify a specific vector in this { 4-Dimension Vector Space } ,
    • namely the vector :
      • { 1, 1, 1, 1 }

        for our use in STEP THREE , next .

  STEP THREE :

  • We take note at this point that :
    • vectors orthogonal to the above-said [ Vector in STEP TWO ]
      • do reside in a { 3-Dimension Sub-Space } .

    We then create the following 4 pairs of vectors :

    • 1st pair :

      { -1, -1, -1, +3 } and { +1, +1, +1, -3 }

    • 2nd pair :

      { -1, -1, +3, -1 } and { +1, +1, -3, +1 }

    • 3rd pair :

      { -1, +3, -1, -1 } and { +1, -3, +1, +1 }

    • 4th pair :

      { +3, -1, -1, -1 } and { -3, +1, +1, +1 }

    And we note here that :

    • First-of-all :
      • the 8 vectors listed-above are orthogonal to the Vector { 1, 1, 1, 1 } and
        • therefore do reside in a { 3-Dimension Sub-Space } ;

    • Secondly :
      • the two (2) vectors within each [ pair of vectors ] above are always complementary :
        • i.e. , they always add up to [ zero ] and always point in the exact-opposite directions .

  STEP FOUR :

  • Let us now join the end-points of each pair of vectors with a straight-line , and
    • these are then our { 4 Symmetric Lines } within a { 3-Dimension Vector Space } .

    And we note here that :

    • the 4 Lines are of equal lengths ,

    • they always intersect one-another at their mid-points , and

    • they always intersect one-another at the exact-same angle .

• And we simply note here the angle of intersection here is [ 70.53 degrees ] , given by :
  • arccos( 1/3 ) = 70.53 degrees

    using of course a [ Vector DOT-PRODUCT ] approach here in our calculations .

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go to the next Section : Section IV - { 5 Symmetric Lines } in a { 4-Dimension Space }

go to the last Section : Section II - { 3 Symmetric Lines } in a { 2-Dimension Plane }

return to the { N+1 Symmetric Lines } HomePage

Original dated 2016-3-22