July 22, 2024
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R2 – Rotation – Simple

This post gives some simple R2 examples of clockwise and counterclockwise rotations around the origin.

Counterclockwise Examples

  1. π^/6 ↺ 1^ = √3/2^ + 1/2i^: The point (1^, 0) on the x-axis rotates by π^/6,  30o^  to the Cartesian point (√3/2^, 1/2i^). That is to say (√3/2^ +1/2i^) algebraically.
  2. π^/4 ↺ 1^ = √2/2^ + √2/2i^: The point (1^, 0) on the x-axis rotates by π^/4, 45o^ to the Cartesian point (√2/2^, √2/2i^). That is to say  (√2/2^ + √2/2i^) algebraically.
  3. π^/3 ↺ 1^ = 1/2^ + √3/2i^: The point (1^, 0) on the x-axis rotates by π^/3, 60o^  to the Cartesian point (1/2^, √3/2i^). That is to say  (1/2^ + √3/2i^) algebraically.
  4. π^/2 ↺ 1^ = i^: The point (1^, 0) on the x-axis rotates by  90o^ or π^/2 to move to the Cartesian point (0,  i^). That is to say  i^ algebraically.
  5. ^/3 ↺ 1^ = 1/2v + √3/2i^: The point (1^, 0) on the x-axis rotates by 2π^/3, 120o^  to the Cartesian point (1/2v, √3/2i^). That is to say  (1/2v + √3/2i^) algebraically.
  6. ^/4 ↺ 1^ = √2/2v + √2/2i^: The point (1^, 0) on the x-axis rotates by 3π^/4, 135o^ to the Cartesian point (√2/2v, √2/2i^). That is to say √2/2v + √2/2i^  algebraically.
  7. ^/6 ↺ 1^ = √3/2v + 1/2i^: The point (1^, 0) on the x-axis rotates by π^/6,  150o^  to the Cartesian point (√3/2v, 1/2i^). That is to say   (√3/2v +1/2i^) algebraically.
  8. π^ ↺ 1^ = 1v: The point (1^, 0) on the x-axis rotates by  180o^ or π^ to move to the Cartesian point (1v, 0). That is to say  1v  algebraically.
  9. ^/6 ↺ 1^ = √3/2v + 1/2iv: The point (1^, 0) on the x-axis rotates by 7π^/6, 210o^  to the Cartesian point (√3/2v, 1/2iv). That is to say (√3/2v + 1/2iv) algebraically.
  10. ^/4 ↺ 1^ = √2/2v + √2/2iv: The point (1^, 0) on the x-axis rotates by 5π^/4, 225o^ to the Cartesian point (√2/2v, √2/2iv). That is to say  √2/2v + √2/2iv algebraically.
  11. ^/3 ↺ 1^ = 1/2v + √3/2iv: The point (1^, 0) on the x-axis rotates by 4π^/3, 240o^  to the Cartesian point (1/2v, √3/2iv). That is to say (1/2v + √3/2iv) algebraically.
  12. ^/2 ↺ 1^ = iv: The point (1^, 0) on the x-axis rotates by  270o^ or 3π^/2 to move to the Cartesian point (0,  iv). That is to say  iv algebraically.
  13. ^/3 ↺ 1^ = 1/2^ + √3/2iv: The point (1^, 0) on the x-axis rotates by 5π^/3, 300o^  to the Cartesian point (1/2^, √3/2iv). That is to say  (1/2^ + √3/2iv) algebraically.
  14. ^/4 ↺ 1^ = √2/2^ + √2/2iv: The point (1^, 0) on the x-axis rotates by 7π^/4, 315o^  to the Cartesian point (√2/2^, √2/2iv) that is to say  (√2/2^ + √2/2iv) algebraically.
  15. 11π^/6 ↺ 1^ = √3/2^ + 1/2iv: The point (1^, 0) on the x-axis rotates by 11π^/6,  330o^  to the Cartesian point (√3/2^, 1/2iv). That is to say   (√3/2^ +1/2iv) algebraically.
  16. ^ ↺ 1^ = 1^: The point (1^, 0) on the x-axis rotates by  360o^ or 2π^ to move back to the Cartesian point (1^, 0). That is to say  1^  algebraically.
  17. π^/2 ↺ 1^ = i^: The point (1^, 0) on the x-axis rotates by π^/2  to move to the Cartesian point (0, i^). That is to say i^ algebraically.
  18. π^/2 ↺ i^ = 1v: The point (0, i^) on the y-axis rotates by π^/2  to move to the Cartesian point (1v, 0). That is to say 1v algebraically.
  19. π^/2 ↺ 1v = iv: The point (1v, 0) on the x-axis rotates by π^/2  to move to the Cartesian point (0, iv). That is to say  iv algebraically.
  20. π^/2 ↺ iv = 1^: The point (0, iv) on the y-axis rotates by π^/2  to move to the Cartesian point (1^, 0). That is to say  1^ algebraically. These last 4 rotations return the point to the beginning 1^.

Other R2 Rotations

A similar set of clockwise R2 rotations can be done using  v values of π. For example: πv/2 ↺ 1^ = iv:

Other R2 rotation values of π^ or πv allow the movement of a point to anywhere on a circle. This circle will have a radius of the distance from the point to the axis point.

Conclusion

Try these examples of R2 rotation with our online calculator.

Next: Cartesian Coordinates and Euler

Previous: Rotation, Flipping and Prop

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