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:
“People are always dying in the Times who don’t seem to die in other papers, and they die at greater length and maybe even with a little more grace.”
—James Reston (b. 1909)
“At length I heard a ragged noise and mirth
Of thieves and murderers: there I him espied
Who straight, Your suit is granted,said, and died.”
—George Herbert (1593–1633)
“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)