Axis–angle representation

In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by three quantities, a unit vector e indicating the direction of an axis of rotation, and an angle θ describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of the unit vector e because its magnitude is constrained. The angle θ scalar multiplied by the unit vector e is the axis-angle vector

The vector itself does not perform rotations, but is used to construct transformations on vectors that correspond to rotations. The rotation occurs in the sense prescribed by the right-hand rule. The rotation axis is sometimes called the Euler axis.

It is one of many rotation formalisms in three dimensions. The axis–angle representation is predicated on Euler's rotation theorem, which dictates that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis.

The axis–angle representation is equivalent to the more concise rotation vector, also called the Euler vector. In this case, both the rotation axis and the angle are represented by a vector codirectional with the rotation axis whose length is the rotation angle θ,

It is used for the exponential and logarithm maps involving this representation.

Say you are standing on the ground and you pick the direction of gravity to be the negative z direction. Then if you turn to your left, you will rotate π/2 radians (or 90°) about the z axis. Viewing the axis-angle representation as an ordered pair, this would be

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