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:
“The grace of novelty and the length of habit, though so very opposite to one another, yet agree in this, that they both alike keep us from discovering the faults of our friends.”
—François, Duc De La Rochefoucauld (1613–1680)
“So much symmetry!
Like the pale angle of time
And eternity.
The great shape labored and fell.”
—N. Scott Momaday (b. 1934)