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:
“I have seen some whose consciences, owing undoubtedly to former indulgence, had grown to be as irritable as spoilt children, and at length gave them no peace. They did not know when to swallow their cud, and their lives of course yielded no milk.”
—Henry David Thoreau (1817–1862)
“So much symmetry!
Like the pale angle of time
And eternity.
The great shape labored and fell.”
—N. Scott Momaday (b. 1934)