R3 – Reversal, Remain and Zero Rules

This post covers the Reversal, Remain and Zero rules of rotation in R3.

The Reversal Rule of Rotation in R3

The third universal rule of rotation in R3 is the Reversal Rule. This rule states that a rotation can be reversed by multiplying the result of the initial rotation by the same operator, but with the sign reversed. In other words, a 90^o turn in one direction can be undone by a 90^o turn in the opposite direction. Even when a point is rotated around its own axis, resulting in no actual displacement, the Reversal Rule still applies.

The following examples show that R3 rotation meets the Reversal Rule:

1^{^} Rotation Followed by Reversal

• 1^{^}*1^{^} = 1^{^} and 1^v*1^{^} = 1^{^}
  • In cases where the rotation of a point is around its own axis, 0 rotation and 0 reversal occurs
• 1^{^}*1^v = 1^v and 1^v*1^v = 1^v
• 1^{^}*i^{^} = j^{^} and 1^v*j^{^} = i^{^}
• 1^{^}*i^v = j^v and 1^v*j^v = i^v
• 1^{^}*j^{^} = i^v and 1^v*i^v = j^{^}
• 1^{^}*j^v = i^{^} and 1^v*i^{^} = j^v

1^v Rotation Followed by Reversal

• 1^v*1^{^} = 1^{^} and 1^{^}*1^{^} = 1^{^}
• 1^v*1^v = 1^v and 1^{^}*1^v = 1^v
• 1^v*i^{^} = j^v and 1^{^}*j^v = i^{^}
• 1^v*i^v = j^{^} and 1^{^}*j^{^} = i^v
• 1^v*j^{^} = i^{^} and 1^{^}*i^{^} = j^{^}
• 1^v*j^v = i^v and 1^{^}*i^v = j^v

i^{^} Rotation Followed by Reversal

• i^{^}*1^{^} = j^v and i^v*j^v = 1^{^}
• i^{^}*1^v = j^{^} and i^v*j^{^} = 1^v
• i^{^}*i^{^} = i^{^} and i^v*i^{^} = i^{^}
• i^{^}*i^v = i^v and i^v*i^v = i^v
• i^{^}*j^{^} = 1^{^} and i^v*1^{^} = j^{^}
• i^{^}*j^v = 1^v and i^v*1^v = j^v

i^v Rotation Followed by Reversal

• i^v*1^{^} = j^{^} and i^{^}*j^{^} = 1^{^}
• i^v*1^v = j^v and i^{^}*j^v = 1^v
• i^v*i^{^} = i^{^} and i^{^}*i^{^} = i^{^}
• i^v*i^v = i^v and i^{^}*i^v = i^v
• i^v*j^{^} = 1^v and i^{^}*1^v = j^{^}
• i^v*j^v = 1^{^} and i^{^}*1^{^} = j^v

j^{^} Rotation Followed by Reversal

• j^{^}*1^{^} = i^{^} and j^v*i^{^} = 1^{^} 
• j^{^}.1^v = i^v and j^v*i^v = 1^v 
• j^{^}*i^{^} = 1^v and j^v*1^v = i^{^} 
• j^{^}*i^v = 1^{^} and j^v*1^{^} = i^v 
• j^{^}*j^{^} = j^{^} and j^v*j^{^} = j^{^} 
• j^{^}*j^v = j^v and j^v*j^v = j^v

j^v Rotation Followed by Reversal

• j^v*1^{^} = i^v and j^{^}*i^v = 1^{^}
• j^v*1^v = i^{^} and j^{^}*i^{^} = 1^v 
• j^v*i^{^} = 1^{^} and j^{^}*1^{^} = i^{^}
• j^v*i^v = 1^v and j^{^}*1^v = i^v 
• j^v*j^{^} = j^{^} and j^{^}*j^{^} = j^{^}
• j^v*j^v = j^v and j^{^}*j^v = j^v

The Remain Rule

The fourth universal rule of rotation in R3 is the Remain Rule. This rule states that a point remains in the same position when it is rotated about its own axis.

The following examples show that R3 rotation meets the Remain Rule:

X-Axis Rotation

• 1^{^}*1^{^} = 1^{^}
• 1^{^}*1^v = 1^v
• 1^v*1^{^} = 1^{^}
• 1^v*1^v = 1^v

Y-Axis Rotation

• i^{^}*i^{^} = i^{^}
• i^{^}*i^v = i^v
• i^v*i^{^} = i^{^}
• i^v*i^v = i^v

Z-Axis Rotation

• j^{^}*j^{^} = j^{^}
• j^{^}*j^v = j^v
• j^v*j^{^} = j^{^}
• j^v*j^v = j^v

Other

The Zero Rules of Rotation in R3

The final universal rule of rotation in R3 is the Zero Rules. The following examples show that R3 rotation meets each case:

Part (a): a * b + a *^–b = 0

• For example:
  • 1^v*i^v + 1^v*i^{^} = j^{^} + j^v = 0
  • j^v*i^{^} + j^v*i^v = 1^{^}+ 1^v = 0

Part (b): a * b + b * a = 0 where a and b are not the same Opposite Type and Sign

• For example:
  • 1^v*i^v + i^v*1^v = j^{^} + j^v = 0
  •  j^v*i^{^} + i^{^}*j^v = 1^{^}+ 1^v = 0

Part (c): a + b(b * a) = 0 where a and b are not the same Opposite Type and Sign

• For example:
  • 1^v + j^v(j^v*1^v) = 1^v + j^v*i^{^} =  1^v + 1^{^} = 0
  • i^{^} + 1^v(1^v*i^{^}) = i^{^} + 1^v*j^v =  i^{^} + i^v = 0

Next: Complex Rotations

Previous: Return and Orthogonal Rules