If we apply two successive reflections to a vector
The angle of this rotation is twice the angle between the reflection vectors
Intuition
One way to think about this property is the following: both reflection and rotations are orthogonal transformations. The difference is that rotation preserves orientation while reflection flips it. If we apply the reflection twice, the first reflection flip the orientation and the second reflection flip it back. Also, if we apply multiple orthogonal transformation, the combined transformation is still orthogonal. Thus, at the end we get an orthogonal transformation that is orientation-preserving, which is a rotation.
Exception
For two reflections around two parallel lines, we get a translation instead:
Related
- rotors construct rotation with this property (and that’s why they use half the angle!)
- The quantum search algorithm is based on this property