What is the Inverse of a Rotation Vector?

The inverse of a rotation vector representation is simply its negative.

A rotation vector is a convenient way to represent a 3D rotation. It's a vector whose direction indicates the axis of rotation and whose magnitude indicates the angle of rotation (often in radians). Think of it as pointing along the axis you spin around, with the length of the vector telling you how much you turn.

Calculating the Inverse

Based on the provided reference, the inverse of a rotation vector m is found by taking its negative. This means reversing the direction of the vector while keeping its magnitude the same.

Mathematically, if Rm(m) represents the rotation corresponding to the vector m, then the inverse rotation Rm(m)⁻¹ is the rotation corresponding to the vector -m.

• Reference Key Point: "the inverse of a rotation vector representation is simply its negative Rm(m)−1=Rm(−m)."

This operation is straightforward:

1. Identify the components of the rotation vector m = [m_x, m_y, m_z].
2. The inverse rotation vector is -m = [-m_x, -m_y, -m_z].

Example:

If a rotation vector is m = [0.5, -1.2, 0.1], representing a rotation about an axis in the direction [0.5, -1.2, 0.1] by an angle equal to the vector's magnitude (sqrt(0.5² + (-1.2)² + 0.1²)), then its inverse rotation vector is simply -m = [-0.5, 1.2, -0.1].

Why is the Inverse the Negative?

Intuitively, performing a rotation R and then performing its inverse R⁻¹ should bring you back to the original orientation. If a rotation vector m causes a specific turn, performing the rotation represented by -m effectively spins you back along the same axis but in the opposite direction by the same amount, perfectly undoing the original rotation.

Consider a simpler axis-angle representation (a, θ), where a is the unit axis vector and θ is the angle. The reference mentions two ways to represent the inverse: (−a, θ) or (a, −θ). A rotation vector m combines these ideas: its direction is the axis a and its magnitude is the angle θ. Negating the vector m effectively negates both the direction of the axis and the sign of the angle simultaneously, achieving the inverse rotation.

The vector negation -m aligns with the (a, −θ) interpretation commonly used when the vector magnitude represents the positive angle.