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, length and/or angle:
“And my spirit is grown to a lordly great compass within,
That the length and the breadth and the sweep of the marshes of
Glynn
Will work me no fear like the fear they have wrought me of yore
When length was failure, and when breadth was but bitterness sore,
And when terror and shrinking and dreary unnamable pain
Drew over me out of the merciless miles of the plain,—
Oh, now, unafraid, I am fain to face
The vast sweet visage of space.”
—Sidney Lanier (1842–1881)
“When at length they rose to go to bed, it struck each man as he followed his neighbour upstairs that the one before him walked very crookedly.”
—R.S. (Robert Smith)
“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)