A rotation of the real plane
around the origin, given the angle
counterclockwise, maps
to
and
to
. Therefore, plane rotations are described in the following way.
A
linear mapping
-
which is given by a
rotation matrix
(with some
)with respect to the standard basis is called
rotation.
A space rotation is a linear mapping of the space
in itself around a rotation axis
(a line through the origin)
with an certain angle
. If the vector
defines the axis, and
and
are orthogonal to
and to each other, and all have length
, then the rotation is described by the matrix
-
with respect to the basis
.