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.
Read more about this topic: Rotation Group SO(3)
Famous quotes containing the words length and/or angle:
“To find the length of an object, we have to perform certain
physical operations. The concept of length is therefore fixed when the operations by which length is measured are fixed: that is, the concept of length involves as much as and nothing more than the set of operations by which length is determined.”
—Percy W. Bridgman (1882–1961)
“I fly in dreams, I know it is my privilege, I do not recall a single situation in dreams when I was unable to fly. To execute every sort of curve and angle with a light impulse, a flying mathematics—that is so distinct a happiness that it has permanently suffused my basic sense of happiness.”
—Friedrich Nietzsche (1844–1900)