Composition of rotational transformations

[seyma]

Show that when basic rotations are combined to find composite rota-
tional transformations, if the rotation is about one of the principal axes of
OXYZ (the fixed frame) the previous resultant rotation matrix is premulti-
pled by the new rotational transformation, and if the rotation is about one
of the principal axes of OUVW (the moving frame) the previous resultant
rotation matrix is postmultipled by the new rotational transformation.

Homework Equations

I know that I can write fixed frame coordinates in terms of moving frame that is the transition matrix which is composed of unit vectors of two coordinates. I call this R. Then I
call fixed frame rotation A and moving frame rotation B. then this becomes A = RB. also for any rotation matrix call R' I can write AR' = R'B.

The Attempt at a Solution

Also I know all transformation matrix is orthonormal. However I cannot know how to prove.

 

[jbsteele]

Let's first consider the rotation about one of the principal axes of OXYZ (the fixed frame). Then the previous resultant rotation matrix is premultiplied by the new rotational transformation. This means that A = RB, where R is the transition matrix composed of the unit vectors of two coordinates and B is the new rotational transformation. To prove this we first use the orthonormality of the matrix. Since both A and R are orthonormal, we have A'A = I, where I is the identity matrix. Similarly we have R'R = I. Using these two equations and applying them to the above equation gives us A'AR' = I. Since A' = A and R' = R, we get AA'R = I. Since A'R = B, we get AB = I. Thus we can conclude that the previous resultant rotation matrix is premultiplied by the new rotational transformation. Now let's consider the rotation about one of the principal axes of OUVW (the moving frame). Then the previous resultant rotation matrix is postmultiplied by the new rotational transformation. This means that AR' = R'B, where R is the transition matrix composed of the unit vectors of two coordinates and B is the new rotational transformation. To prove this we first use the orthonormality of the matrix. Since both A and R are orthonormal, we have A'A = I, where I is the identity matrix. Similarly we have R'R = I. Using these two equations and applying them to the above equation gives us A'AR' = I. Since A' = A and R' = R, we get A'AR' = I. Since A'R = B, we get AR' = B. Thus we can conclude that the previous resultant rotation matrix is postmultiplied by the new rotational transformation.