A vector in space has to be expressed using basis as a reference and coordinate , such that,

A transform of a vector is done by

This transform can be view as

The first one is change the basis and keep the coordinate, and the later is change the coordinate and keep the basis.

The Euler angle, defined as,

Since the rotational matrix is fixed and not based on any basis, how to understand this transform can be viewed differently.

The implementation of the rotation usually like this:

And the rotation order is . In this order, the rotation is using body (or current, intrinsic) axis.

If the rotation order is , the rotation is using fixed (or global, external, Laboratory) axis.

To understand this, we have to add back the basis . When using the body axis, the left most matrix actually act on the basis (operationally)

When using the fixed axis,

Thus, if we using Euler angle in fixed frame, the correct matrix is

In the following, we use fixed frame, unless specified.

Since using Euler angle in changing frame, the rotation is fixed on the new axis . However, the matrix

is NOT a rotation on the axis with the angle. As we can see that, for a zero rotation , any vector will still be transformed by .

Now, suppose we have a rotation axis and an rotation angle , the rotation matrix can be constructed as follow.

Notice that the rotational matrix is a unitary transform, its eigen-vectors are ,

Using , solve .

Since the eigen vector of are

Since , thus,

such that

Notice that the corresponding eigen vector for the eigen value is .

Then the rotational matrix is

Another to look at this problem is that, an operator, can be “decomposed” into its eigen vectors and eigen-values, using Dirac notation,

The rotation round the vector can be transformed use from , Thus, the vector has to be transform, so the operator

This is a relation between fixed frame rotation to a body rotation! can be the body z-axis, y-axis, or x-axis in particular.

For example, Let be the rotated y-axis, such that

Notice that

The above formula are not difficult to imagine, decompose the middle operator into the eigen-system and use Dirac notation. For example, the rotate of the z-axis around y-axis with 90 degree is the x-axis. Thus,

We recovered the relation of the Lab frame to body frame for y-axis!

Also, given a vector rotated by to around unit vector is