Rotation:
In this post, we delve into the Return and Orthogonal rules of rotation within the three-dimensional space R3. These principles are foundational to understanding the universal rules governing rotations, especially in contexts like Wave Numbers, where rotation operations play a crucial role.
A rotation in R3 involves moving a point from one position to another, with both positions represented by pairs of Opposite Values. This movement occurs in a circular path around a fixed axis, also defined by a pair of Opposite Values. Throughout the rotation, the axis remains stationary, ensuring the consistency of the rotational motion. These concepts are grounded in the axioms that support the mathematical framework of rotations.
The Return Rule
The first rule of rotation in R3 is the Return Rule. It states that continuously rotating a point by the same angle will return the point to its original position 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.
The following examples show that R3 rotation meets the Return Rule:
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^)
- 2π^/3 ↺(4^+ 5i^ + 6j^) (1^ + 2i^ + 3j^) = (2.28958^ + 1.52473i^ + 2.53634j^)
- 2π^/3 ↺(4^+ 5i^ + 6j^) (2.28958^ + 1.52473i^ + 2.53634j^) = (1.69743^ + 2.70904i^ + 1.94419j^)
- 2π^/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 rotation by a unitary value is 90o. Rotating by an Opposite Value with a ^ sign indicates a counterclockwise direction, while rotating by an Opposite Value with a v sign indicates a clockwise direction. This rule is fully represented in the R3 multiplication table.
Next: Reversal, Remain and Other Rules
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