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 man who, from the beginning of his life, has been bathed at length in the soft atmosphere of a woman, in the smell of her hands, of her bosom, of her knees, of her hair, of her supple and floating clothes, ... has contracted from this contact a tender skin and a distinct accent, a kind of androgyny without which the harshest and most masculine genius remains, as far as perfection in art is concerned, an incomplete being.”
—Charles Baudelaire (1821–1867)
“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)