Maths & numbers
Quaternion vector rotation
Normalise a Hamilton quaternion and use it to rotate a 3D vector.
A clear answer. Here’s how.
Treats quaternion components in [w,x,y,z] Hamilton order and normalises the quaternion before applying active right-handed rotation q·[0,v]·conjugate(q). Zero quaternion is rejected. q and −q represent the same rotation. Reports the normalised quaternion and rotated coordinates; vector length is preserved within floating-point rounding. Component ordering and handedness may differ in other software.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Vector X
- 1
- Vector Y
- 0
- Vector Z
- 0
- Quaternion real component w
- 0.7071067811865476
- Quaternion X component
- 0
- Quaternion Y component
- 0
- Quaternion Z component
- 0.7071067811865476
Example result
- Rotated X
- 0.000000000000000222044604925
- Rotated Y
- 1
- Rotated Z
- 0
- Normalised quaternion [w,x,y,z]
- 0.7071067811865475, 0, 0, 0.7071067811865475
Hamilton quaternion, normalised before active right-handed rotation q·[0,v]·conjugate(q). Zero quaternion is rejected.