Unit Vectors

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 XVI - Summarizing our Findings on the 6 Regions

Summary for the Section :

In this Section , we shall :
- set up a set of { 2 orthogonal [ unit vectors ] } in the X-and-Y directions ,
  - for our use here and in the next Section ;
  and
- summarize our findings in the last 6 Sections and do some further analysis .

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Recalling [ R ] / [ RHO ] / [ THETA ] :

Let us recall that we did set up [ R ] / [ RHO ] / [ THETA ] in Section IX ,
- with :
  - [ R ] being the radius of the Circle in the { Reference Structure } ,
    
    and
  - [ RHO ] and [ THETA ] being the 2 variables fully defining the position of the seed-point [ Point H ] .
And we can then establish the positions [ Point A ] , [ Point B ] and [ Point C ] ,
- via the { Standardized Construction Procedure } detailed back in Section VII ;
  - as per the set of 3 diagrams above .

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Establishing [ Unit Vectors ] in the X-direction and the Y-direction :

For our analysis to follow , it is necessary for us to set up :
- a set of { 2 orthogonal [ unit vectors ] } in the X-direction and the Y-directions ,
  - i.e. : the vectors [ e-sub-X ] and [ e-sub-Y ] respectively ,
    - as shown in the diagram on-the-left below .
We then also establish an additional 3 [ unit vectors ] :
- [ Vector e-sub-T ] pointing outwards and being orthogonal to [ Line D-E ] ,
- [ Vector e-sub-U ] pointing outwards and being orthogonal to [ Line E-F ] , and
- [ Vector e-sub-V ] pointing outwards and being orthogonal to [ Line D-E ] ;
  
  as shown in the diagram on-the-right above .
We then have :

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Our Findings in the last 6 Sections :

In the last 6 Sections , we were able to establish these 3 expressions for :
- the [ length of Line Segment H-A ] , [ length of Line Seqgemnt H-B ] and [ length of Line Segment H-C ] ;
  
  as follows :
But an extra word of caution here :
- When [ Point H ] happens to lie in { Region Aaron } ,
  - the first expression above will yield a negative value for [ length of Line Segment H-A ] ,
    
    and
    
    this will simply indicate that the [ Vector H-A ] will be pointing in the opposite direction of [ Vector e-sub-T ] .
- When [ Point H ] happens to lie in { Region Baron } ,
  - the second expression above will yield a negative value for [ length of Line Segment H-B ] ,
    
    and
    
    this will simply indicate that the [ Vector H-B ] will be pointing in the opposite direction of [ Vector e-sub-U ] .
- when [ Point H ] happens to lie in { Region Cyron } ,
  - the third expression above will yield a negative value for [ length of Line Segment H-C ] ,
    
    and
    
    this will simply indicate that the [ Vector H-C ] will be pointing in the opposite direction of [ Vector e-sub-V ] .
So , no problems here .
And of course :
- When [ Point H ] happens to lie on [ Line D-E ] :
  - [ distance H-A ] will be [ zero ] ,
    - because [ Point H ] and [ Point A ] will be the same point ;
    and the first expression above for the [ length of Line Segment H-A ] will also yield a [ zero ] value .
- When [ Point H ] happens to lie on [ Line E-F ] :
  - [ distance H-B ] will be [ zero ] ,
    - because [ Point H ] and [ Point B ] will be the same point ;
    and the second expression above for the [ length of Line Segment H-B ] will also yield a [ zero ] value .
- When [ Point H ] happens to lie on [ Line F-D ] :
  - [ distance H-C ] will be [ zero ] ,
  and the third expression above for the [ length of Line Segment H-C ] will also yield a [ zero ] value .
So, no problems here either .

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The Position Vectors for [ Point A ] / [ Point B ] / [ Point C ] :

Let us now take a quick look at the position vectors for [ Point A ] , [ Point B ] , and [ Point C ] ;
- as shown in the diagram on-the-left below .
We then have :
- [ Vector O-A ] = [ Vector O-H ] + [ Vector H-A ]
- [ Vector O-B ] = [ Vector O-H ] + [ Vector H-B ]
- [ Vector O-C ] = [ Vector O-H ] + [ Vector H-C ]
  
  as shown above .
And for [ Vector O-H ] , we have :
And for the 3 vectors [ Vector H-A ] / [ Vector H-B ] / [ Vector H-C ] , we have :
Let us bring in from above these values we have established in the last 6 Sections :
Substituting these into the 3 equations immediately preceeding , we can now write :
And on further expansion , we have :
Let us recall that :
And based on this and the 3 equations immediately preceeding :
- For the expression :
  - [ Vector O-A ] = [ Vector O-H ] + [ Vector H-A ]
  we can now write :
- For the expression :
  - [ Vector O-B ] = [ Vector O-H ] + [ Vector H-B ]
  we can now write :
- For the expression :
  - [ Vector O-C ] = [ Vector O-H ] + [ Vector H-C ]
  we can now write :