The inverse of a rotation vector is simply its negative.
In more detail:
A rotation vector represents a rotation using a vector whose direction is the axis of rotation and whose magnitude is the angle of rotation in radians. If we represent a rotation as a rotation vector r, then its inverse, which undoes the original rotation, is -r.
Think of it this way:
- The rotation vector r = (a, θ) represents a rotation of angle θ around axis a.
- The inverse rotation -r = (-a, θ) or (a, -θ) represents a rotation of the same angle but in the opposite direction or around the opposite direction of the axis. Rotating by θ and then by -θ results in no net rotation.
Mathematically, if R(m) represents the rotation matrix generated from the rotation vector m, then the inverse of the rotation matrix, R(m)-1, is equal to the rotation matrix generated from the negative of the rotation vector, R(-m). That is:
R(m)-1 = R(-m)
Example:
Let's say you have a rotation vector v = (1, 0, 0) π/2. This represents a 90-degree rotation around the x-axis. The inverse of this rotation is -v = (-1, 0, 0) π/2, or equivalently, *(1, 0, 0) -π/2*. This represents a -90-degree rotation around the x-axis (or a 90-degree rotation around the negative* x-axis), which effectively undoes the original rotation.
In summary, finding the inverse of a rotation vector is a straightforward process of negating its components. This negative vector represents the rotation that will precisely reverse the original rotation.
