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:
“Punishment followed on a grand scale. For ten days, an unconscionable length of time, my father blessed the palms of his child’s outstretched, four-year-old hands with a sharp switch. Seven strokes a day on each hand; that makes one hundred forty strokes and then some. This put an end to the child’s innocence.”
—Christoph Meckel (20th century)
“The inhabitants of earth behold commonly but the dark and shadowy under side of heaven’s pavement; it is only when seen at a favorable angle in the horizon, morning or evening, that some faint streaks of the rich lining of the clouds are revealed.”
—Henry David Thoreau (1817–1862)