Tait-Bryan Angles
This post looks at simple R3 rotations that consist of 90o rotations around a single axis on the unit sphere. A 90^o axis-angle rotation around the x-axis is a z-y’-x’’ rotation using the Tait-Bryan angles of φ = 90^o, θ = 0o and ψ= 0o. Rotation of 90^o around the x-axis moves the point Y^ to the location of Z^, and Z^ to Yv. As a result, the xy plane now intersects the YZ plane orthogonally through the y-axis to form the line of nodes called N. φ is the angle between the N line and the Y-axis which is 90^o. This is the angle through which the Y^ point was rotated around the x-axis.
Unitary R3 Rotations
All the simple rotations looked at are rotations around either the x, y or z-axes and so have only one Tait-Bryan angle of 90o, the others being 0. The Orthogonal Rule from the Axioms of Rotation states that multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90^o. So, all the simple R3 rotations are the equivalent of multiplying by one of the unitaries 1^, 1v, i^, iv, j^ or jv.
Try out these rotations using the online calculator.
90o Rotations from (1^, 0, 0)
- π^/2 ↺z (1^,0, 0) = i^ or in multiplication j^*1^ = i^.
- The point (1^,0, 0) on the x-axis rotates by 90^o counterclockwise or π^/2 around the z-axis. So, it moves to the point (0, i^,0), or i^ algebraically.
- π^/2 ↺y (1^,0, 0) = jv or in multiplication i^*1^ = jv.
- The point (1^,0, 0) on the x-axis rotates by 90^o or π^/2 around the y-axis. So, it moves to the point (0, 0, jv), or jv algebraically.
- π^/2 ↺x (1^,0, 0) = 1^or in multiplication 1^*1^ = 1^.
- The point (1^,0,0) on the x-axis rotates by 90^o counterclockwise or π^/2 around the x-axis but as the point is already on the x-axis, it stays at the point (1^,0, 0), or 1^ algebraically.
- πv/2 ↺z (1^,0, 0) = iv or in multiplication jv*1^ = iv.
- The point (1^,0, 0) on the x-axis rotates by 90vo clockwise or πv/2 around the z-axis. So, it moves to the point (0, iv, 0), or iv algebraically.
- πv/2 ↺y (1^,0, 0) = j^ or in multiplication iv*1^ = j^.
- The point (1^,0,0) on the x-axis rotates by 90vo clockwise or πv/2 around the y-axis. So, it moves to the point (0, 0, j^), or j^ algebraically.
- πv/2 ↺x (1^,0, 0) = 1^ or in multiplication 1v*1^ = 1^.
- The point (1^,0, 0) on the x-axis rotates by 90vo clockwise or πv/2 around the x-axis but as the point is already on the x-axis, it stays at the point (1^,0, 0), or 1^ algebraically.
- πv/2 ↺z (2^,3iv, 4jv) = (3v, 2iv, 4jv) or in multiplication jv * (2^ + 3iv + 4jv) = (3v + 2iv + 4jv).
- The point (2^,3iv, 4jv) rotates by 90vo clockwise or πv/2 around the z-axis. So, it moves to the point (3v, 2iv, 4jv), or (3v + 2iv + 4jv) algebraically.
Other Possible 90o or π?/2 rotations
| Start Point | Rotation | End Point | Algebra | |
| 1 | (1v,0,0) | π^/2↺z | (0, iv,0) | j^*1v = iv |
| 2 | (1v,0,0) | π^/2↺y | (0, 0, j^) | i^*1v = j^ |
| 3 | (1v,0,0) | π^/2↺x | (1v,0,0) | 1^*1v = 1v |
| 4 | (1v,0,0) | πv/2↺z | (0, i^,0) | jv*1v = i^ |
| 5 | (1v,0,0) | πv/2↺y | (0, 0, jv) | iv*1v = jv |
| 6 | (1v,0,0) | πv/2↺x | (1v, 0, 0) | 1v*1v = 1v |
| 7 | (0, i^,0) | π^/2↺z | (1v,0,0) | j^*i^ = 1v |
| 8 | (0, i^,0) | π^/2↺y | (0, i^,0) | i^*i^ = i^ |
| 9 | (0, i^,0) | π^/2↺x | (0, 0 , j^) | 1^*i^ = j^ |
| 10 | (0, i^,0) | πv/2↺z | (1^,0,0) | jv*i^ = 1^ |
| 11 | (0, i^,0) | πv/2↺y | (0, i^,0) | iv*i^ = i^ |
| 12 | (0, i^,0) | πv/2↺x | (0, 0 , jv) | 1v*i^ = jv |
| 13 | (0, iv,0) | π^/2↺z | (1^,0,0) | j^*iv = 1^ |
| 14 | (0, iv,0) | π^/2↺y | (0, iv,0) | i^*iv = iv |
| 15 | (0, iv,0) | π^/2↺x | (0, 0, jv) | 1^*iv = jv |
| 16 | (0, iv,0) | πv/2↺z | (1v,0,0) | jv*iv = 1v |
| 17 | (0, iv,0) | πv/2↺y | (0, iv,0) | iv*iv = iv |
| 18 | (0, iv,0) | πv/2↺x | (0, 0, j^) | 1v*iv = j^ |
| 19 | (0, 0, j^) | π^/2↺z | (0, 0, j^) | j^*j^ = j^ |
| 20 | (0, 0, j^) | π^/2↺y | (1^,0,0) | i^*j^ = 1^ |
| 21 | (0, 0, j^) | π^/2↺x | (0, iv,0) | 1^*j^ = iv |
| 22 | (0, 0, j^) | πv/2↺z | (0, 0, j^) | jv*j^ = j^ |
| 23 | (0, 0, j^) | πv/2↺y | (1v,0,0) | iv*j^ = 1v |
| 24 | (0, 0, j^) | πv/2↺x | (0, i^,0) | 1v*j^ = i^ |
| 25 | (0, 0, jv) | π^/2↺z | (0, 0, jv) | j^*jv = jv |
| 26 | (0, 0, jv) | π^/2↺y | (1v, 0, 0) | i^*jv = 1v |
| 27 | (0, 0, jv) | π^/2↺x | (0, i^,0) | 1^*jv = i^ |
| 28 | (0, 0, jv) | πv/2↺z | (0, 0, jv) | jv*jv = jv |
| 29 | (0, 0, jv) | πv/2↺y | (1^,0, 0) | iv*jv = 1^ |
| 30 | (0, 0, jv) | πv/2↺x | (0, iv, 0) | 1v*jv = iv |
Next: Multiplication Table
Previous: Euler and Tait-Bryan Angles