Rotation matrix

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A rotation matrix is a tensor which rotates one Cartesian coordinate system into another.

For example, the rotation matrix for rotating by the angle (right-hand rule) about the axis is:

With this matrix, a vector , referred to one coordinate system, may be referred to another coordinate system, rotated from the first by the angle . In the new coordinate system, the same quantity has vector components given by

The rotation matrix for rotating by the angle about the axis is:

If the coordinate system is rotated further, by an angle about the new -axis, the same quantity has vector components given by

Finally, rotation matrix for rotating by the angle about the axis is:

Any rotation can thus be constructed out of these primitive rotations, about coordinate axes.

See also Tensor algebra.