July 22, 2024
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R3 – Return and Orthogonal Rules

Standard Rotation Definition:

This post addresses the Return and Orthogonal rules of rotation in R3. These are the first two of the universal rules of rotation. The rotation operation is a fundamental operation in Wave Numbers and therefore is supported by the facts expressed in its axioms. A rotation in R3 moves a point represented by Opposite Values to a location represented by other Opposite Values. It is a circular movement around  an axis which is represented by a pair of Opposite Values. The axis of rotation remains fixed during rotation.

The Return Rule

The first rule of rotation in R3 is the Return Rule. It states that continuous rotation of a point by the same rotation returns a point to its original location when the total rotation is a multiple of 360o.  Consequently, all other axes are reachable by continuous rotation of a point by some rotation. 

These rotations routes are called Return Rings. For example:

1^ Multiplication in R3

  • 1^*1^ = 1^
  • 1^*1v = 1v
  • 1^*i^ = j^, 1^*j^ = iv, 1^*iv = jv, 1^*jv = i^
  • 1^*iv = jv, 1^*jv = i^, 1^*i^ = j^, 1^*j^ = iv
  • 1^*j^ = iv, 1^*iv = jv, 1^*jv = i^, 1^*i^ = j^
  • 1^*jv = i^, 1^*i^ = j^, 1^*j^ = iv, 1^*iv = jv

1v Multiplication in R3

  • 1v*1^ = 1^
  • 1v*1v = 1v
  • 1v*i^ = jv, 1v*jv = iv, 1v*iv = j^, 1v*j^ = i^
  • 1v*iv = j^ 1v*j^ = i^, 1v*i^ = jv, 1v*jv = iv
  • 1v*j^ = i^, 1v*i^ = jv, 1v*jv = iv, 1v*iv = j^
  • 1v*jv = iv, 1v*iv = j^, 1v*j^ = i^, 1v*i^ = jv

i^ Multiplication in R3

  • i^*1^ = jv, i^*jv = 1v, i^*1v = j^, i^*j^ = 1^
  • i^*1v = j^, i^*j^ = 1^, i^*1^ = jv, i^*jv = 1v
  • i^*i^ = i^
  • i^*iv = iv
  • i^*j^ = 1^, i^*1^ = jv, i^*jv = 1v, i^*1v = j^
  • i^*jv = 1v, i^*1v = j^, 1^*j^ = 1^, i^*1^ = jv

Other Multiplications in R3

  • 1^*(1^ + i^ + j^) = (1^ + j^ + iv)
    • 1^*(1^ + j^ + iv) = (1^ + iv + jv)
    • 1^*(1^ + iv + jv) = (1^ + jv + i^)  
    • 1^*(1^ + jv + i^) = (1^ + i^ + j^)

  • ^/3 ↺(4^+ 5i^ + 6j^) (1^ + 2i^ + 3j^) = (2.28958^ + 1.52473i^ + 2.53634j^
    • ^/3 ↺(4^+ 5i^ + 6j^)  (2.28958^ + 1.52473i^ + 2.53634j^) = (1.69743^ + 2.70904i^ + 1.94419j^)
    • ^/3 ↺(4^+ 5i^ + 6j^)  (1.69743^ + 2.70904i^ + 1.94419j^) = (1^ + 2i^ + 3j^)

Try these examples of R3 rotation with our online calculator.

The Orthogonal rule of Rotation in R3

The second rule of rotation in R3 is the Orthogonal rule. It states that the rotation by a unitary is 90o.  Rotation by an Opposite Value with a ^ sign is in a counterclockwise direction. In contrast, rotation by an Opposite Value with a v sign is in a clockwise direction.

Next: Reversal, Remain and Other Rules

Previous: Multiplication Table

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