On Breaking-down any Triangle into
a pair of Mirror-Imaged Equilateral Triangles

by Frank C. Fung ( 1st published in September, 2021. )

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Section L - Confirming the { FENG-Line } as a Principal Axes

Summary for the Section :

• In this Section , we shall confirm that :
  • the { FENG-Line } is indeed a Principal Axes of Rotation for a { 3-Equal-Mass System } of [ Mass-A ] / [ Mass-B ] / Mass-C ] .

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A Rather Preculiar Analytic Procedure :

• Let us now bring in a rather preculiar/off-track analytic procedure .

  FIRST :

  • We set up a { set of 2 orthogonal-axis } on a { 2-Dimensional Plane } , namely :
    • the { P-Q Axis } as shown in the diagram on-the-left below .

    Let us also set up the 2 corresponding { unit-vectors } in the { P-direction } and in the { Q-diection } , namely :

    • the unit-vector [ Vector e-sub-P ] and the unit-vector [ Vector e-sub-Q ] respectively ;
      • as per the diagram in-the-middle below .

  SECONDLY :

  • Let us bring in a random fixed-value for our use to follow immediately , namely :
    • the fixed-value [ 2 times THETA-sub-F ] being the angle as marked-off in magenta color ;
      • in the diagram on-the-right below .

  

  

• And based on this fixed-value of [ 2 times THETA-sub-F ] ,
  • we can then set up a set of 3 unit-vectors namely :
    • unit-vector [ Vector e-sub-V1 ] ,

    • unit-vector [ Vector e-sub-V2 ] , and

    • unit-vector [ Vector e-sub-V3 ] ;

    so that :

    • 

    • 

    • 

      as per the set of 4 diagrams above .

  And we take note at this point that :

  • the said 3 said [ unit-vectors ] are tri-symmetric in nature ,
    • and that is to say :
      • each [ unit-vector ] points in a direction differing from another by the angle of [ 120 degrees ] / [ 2-times-PI-over-3 ] .

  As such , the sum of the 3 said [ unit-vectors ] adds up to [ zero ] , i.e. :

  • 

  And because of this ,

  • the projections of these 3 [ unit-vectors ] onto any line passing thru the Point-Of-Origin [ Point O ]
    • will always add up to [ zero ] .

• In particular ,
  • for the sum of the projections onto the [ P-Axis ] , we have :
    • 

  • for the sum of the projections onto the [ Q-Axis ] , we have :
    • 

  And we take note here that :

  • 

  • 

  Therefore :

  • 

  • 

• Let us bring back these 2 equations from above :
  • 

  • 

  and substitute therein the results immediately preceeding to yield :

  • 

  • 

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Confirming the { FENG-Line } as a Principal Axes :

• Let us now bring back the expression for [ Lamda X-Y ] from the last Section - Section L :
  • 

  Applying the 2nd of the 2 results immediately preceeding then yields us this :

  • 

  Therefore :

  • 

  This then confirms that :

  • the { FENG-Line } is indeed a Principal Axes for a { 3 Equal-Mass System } .

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What's next :

• Let us now take a quick look at the { Equal-Mass Three-Body Problem } , next .

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go to the next Section : Section LI - Special Comments on the { Equal-Mass Three-Body Problem }

go to the last Section : Section XLIX - Principal Axes for a { 3-Equal-Mass System }

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07