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)
“What though the traveler tell us of the ruins of Egypt, are we so sick or idle that we must sacrifice our America and today to some man’s ill-remembered and indolent story? Carnac and Luxor are but names, or if their skeletons remain, still more desert sand and at length a wave of the Mediterranean Sea are needed to wash away the filth that attaches to their grandeur. Carnac! Carnac! here is Carnac for me. I behold the columns of a larger
and purer temple.”
—Henry David Thoreau (1817–1862)
“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)