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A2.5 The 3×3 TransformationMatrix Notation

To run the examples in this section, run the initialization cells in Section A2.1 to create the Axes graphics. All rotations may be described using a 3×3 matrix. In reality, the other forms that we have discussed so far are internally converted into this rotation matrix.

The normal 3×3 unrotated matrix is the identity matrix.



Any 3×3 matrix should be orthogonal and have an inverse. Scaling is ignored. The matrix is orthonormalized before being used. A matrix that cannot be orthonormalized is ignored.

A useful way to think of the 3×3 matrix is to remember that each column encodes the x, y, and z directions of the object. Here is an example.


This shows the matrix using input form.


The object's x axis points along the {1, -1, 0} direction. In Dynamic Visualizer, the normal viewpoint has x increases to the right, y increases away from the camera, and z increases upward. For example, you can rotate the object 45 degrees about the z axis and preserve the right-handedness of the coordinate system.



Here is the general form of a rotation about the z axis.



Only the direction of each vector component is important, so {1, -1, 0} is equivalent to the normalized vector . Please note that an easier method of obtaining a rotation about a given axis is to use the AxisAndAngle notation. For example, to achieve the same orientation as the TransformationMatrix presented previously, you can use the following.