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)
“Men sometimes speak as if the study of the classics would at length make way for more modern and practical studies; but the adventurous student will always study classics, in whatever language they may be written and however ancient they may be. For what are the classics but the noblest recorded thoughts of man?... We might as well omit to study Nature because she is old.”
—Henry David Thoreau (1817–1862)
“It is a mistake, to think the same thing affects both sight and touch. If the same angle or square, which is the object of touch, be also the object of vision, what should hinder the blind man, at first sight, from knowing it?”
—George Berkeley (1685–1753)