Rotation Group SO(3) - Length and Angle

Length and Angle

Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length:

It follows that any length-preserving transformation in R3 preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on R3. This is equivalent to requiring them to preserve length.