July 22, 2024
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R2 – Reversal, Remain and Zero Rules

This post describes 3 more universal rules of rotation in R2, the Reversal, Remain and Zero rules.

The Reversal Rule in R2

The third universal rotation rule that applies in R2 is the Reversal rule . A rotation reverses when the result of the first rotation is multiplied by the Operator of the first computation with the sign reversed. In other words, a 90o degree turn in one direction can be reversed by turning back in the other direction.   The Reversal Rule does not apply to R1 and only applies partially to R2 because a free axis of rotation is not available. The lack of a free axis results in no rotation when multiplying by 1^ and 180o rotation when multiplying by 1v.

Reversal Rule and Multiplication by 1^ or 1v

The Reversal Rule does not apply to multiplication by 1^ or 1v.

  • 1^*1^ = 1^; 1v*1^=1v
  • 1^*1v = 1v; 1v*1v= 1^
  • 1^*i^ = i^; 1v*i^= iv
  • 1^*iv = iv; 1v*iv= i^
  • 1v*1^ = 1v; 1^*1v=1v
  • 1v*1v = 1^; 1^*1^= 1^
  • 1v*i^ = iv; 1^*iv= iv
  • 1v*iv = i^; 1^*i^= i^

Reversal Rule and Multiplication by i^ or iv

The Reversal Rule applies to all multiplications by of i^ and iv.

  • i^*1^ = i^; iv*i^=1^
  • i^*1v = iv; iv*iv= 1v
  • i^*i^ = 1v; iv*1v= i^
  • i^*iv = 1^; iv*1^= iv
  • iv*1^ = iv; i^*iv=1^
  • iv*1v = i^; i^*i^= 1v
  • iv*i^ = 1^; i^*1^= i^
  • iv*iv = 1v; i^*1v = iv

The Remain Rule in R2

The fourth universal rule of rotation that applies in R2 is the Remain Rule. A point remains in the same place when rotated about its own single point axis. For example:

  • π^/4 ↺(0 ,1v) (0, 1v) = (0, 1v)
  • π^/2 ↺(0 ,iv) (0, iv) = (0, iv)
  • π^/2 ↺(1^,iv) (1^,iv) = (1^,iv)

Unlike with R3, this does not transfer to unitary multiplication because there is no free axis of rotation. For example: iv*iv = 1v.

The Zero Rule in R2

The Zero Rule only applies to R3.


Try these examples of R2 rotation and multiplication with our online calculator.

Next: Rotation and Multiplication Examples

Previous: Return and Orthogonal Rules

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