November 2, 2024
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R2 – Rotation – Multiplication Table

Orthogonal Rule used to Derive Multiplication Table

The Axioms of Rotation provide the Orthogonal rule that states that multiplication by a unitary is the equivalent of rotation around the unitary’s single point axis by 90o. This allows the derivation of the R2 unitary multiplication table that follows.

Multiplication by i^

Using the post-it note prop, it is easy to see that a counterclockwise rotation of 90^o around the imaginary z-axis moves a point located at (1^, 0) to (0, i^). The Orthogonal rule defines this as multiplication by i^, i.e. i^*1^ = i^. Furthermore, another rotation of 90^o around the imaginary z-axis moves a point located at (0, i^) to (1v, 0). Another rotation of 90^o moves a point located at (1v, 0) to located at (0, iv). A final rotation of 90^o moves a point located at (iv, 0) to the location (1^, 0).

This application of rotation by i^ gives the third row of the R2 multiplication table rules:

i^*1^ = i^,   i^*1v = iv,   i^*i^= 1v,   i^*iv = 1^

Multiplication by iv

Using the post-it note prop, it is easy to see that a clockwise rotation of 90vo around the imaginary z-axis moves a point located at (1^, 0) to (0, iv). The Orthogonal rule defines this as multiplication by iv, i.e. iv*1^ = iv. Further clockwise rotation of 90vo by iv move the point located at (0, iv) first to (1v, 0), secondly to (0, i^) and finally to (1^, 0).

This application of rotation by iv gives the fourth row of the R2 multiplication table rules:

iv*1^ = iv,   iv*1v = i^,    iv*i^= 1^,   iv*iv = 1v

Multiplication by 1^

The R2 rules for multiplication by 1^ can be derived for the table from the rules for i^ and iv as follows:

Given:    1^ = iv*i^ or i^*iv

  • => 1^*1^= iv*i^*1^ = iv*i^ = 1^ or
  • => 1^*1^ = i^*iv*1^ = i^*iv = 1^
    • so 1^*1^ = 1^ 
  • => 1^*i^ = iv*i^*i^ = iv*1v =  i^  or
  • => 1^*i^ = i^*iv*i^ = i^*1^ = i^
    •  so 1^*i^ =  i^ 
  • => 1^*1v = iv*i^*1v = iv*iv =  1v or
  • => 1^*1v = i^*iv*1v = i^*i^ = 1v
    • so 1^*1v =  1v   
  • => 1^*iv = iv*i^*iv = iv*1^ =  iv or
  • => 1^*iv = i^*iv*iv = i^*1v = iv
    • so 1^*iv =  iv 

This gives the first row of the R2 multiplication table rules:

1^*1^ = 1^,  1^*1v =1v,  1^*i^= i^,  1^*iv = iv   

Multiplication by 1v

Finally, the R2 rules for multiplication by 1v can also be derived for the table from the rules for i^ and iv as follows:

Given:    1v = iv*iv or i^*i^

  • => 1v*1^ = iv*iv*1^ = iv*iv = 1v or
  • => 1v*1^ = i^*i^*1^ = i^*i^ = 1v
    • so 1v*1^ = 1v 
  • => 1v*i^ = iv*iv*i^ = iv*1^ =  iv or
  • => 1v*i^ = i^*i^*i^ = i^*1v = iv
    • so 1v*i^=  iv   
  • => 1v*1v = iv*iv*1v = iv*i^ =  1^ or
  • =>1v*1v = i^*i^*1v = i^*iv = 1^
    • so  1v*1v = 1^
  • => 1v*iv = iv*iv*iv = iv*1v =  i^or
  • =>1v*iv = i^*i^*iv = i^*1^ = i^
    • so  1v*iv =  i^ 


R2 Multiplication Table

Putting all this together gives the R2 unitary multiplication table.

*

1^

1v

i^

iv

1^

1^

1v

i^

iv

1v

1v

1^

iv

i^

i^

i^

iv

1v

1^

iv

iv

i^

1^

1v

Conclusion

Try these examples of R2 multiplication with our online calculator.

Next: Universal Rules of Rotation

Previous: Cartesian Coordinates and Euler

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