vectors by an angle about the -, -, or -axis, in three dimensions, using the right-hand rule—which codifies their alternating signs. Notice that the right-hand rule only works when multiplying . (The same matrices can also represent a clockwise rotation of the axes.) For column vectors, each of these basic vector rotations appears counterclockwise when the axis about which they occur points toward the observer, the coordinate system is right-handed, and the angle is positive. , for instance, would rotate toward the a vector aligned with the , as can easily be checked by operating with on the vector : This is similar to the rotation produced by the above-mentioned two-dimensional rotation matrix. See below for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices. General rotations Other rotation matrices can be obtained from these three using matrix multiplication. For example, the product represents a rotation whose yaw, pitch, and roll angles are , and , respectively. More formally, it is an intrinsic rotation whose Tait–Bryan angles are , , , about axes , , , respectively. Similarly, the product represents an extrinsic rotation whose (improper) Euler angles are , , , about axes , , . These matrices produce the desired effect only if they are used to premultiply column vectors, and (since in general matrix multiplication is not commutative) only if they are applied in the specified order (see Ambiguities for more details). The order of rotation operations is from right to left; the matrix adjacent to the column vector is the first to be applied, and then the one to the left. Conversion from rotation matrix to axis–angle Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis (Euler rotation theorem). There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation). Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix. Determining the axis Given a rotation matrix , a vector parallel to the rotation axis must satisfy since the rotation of around the rotation axis must result in . The equation above may be solved for which is