Composition of rotational transformations

In summary, when combining basic rotations to find composite rotational transformations, if the rotation is about one of the principal axes of the fixed frame OXYZ, the previous resultant rotation matrix is premultiplied by the new rotational transformation. Similarly, if the rotation is about one of the principal axes of the moving frame OUVW, the previous resultant rotation matrix is postmultiplied by the new rotational transformation. This can be represented by the equations AA' = RA' = R'B and AR' = RB' = R'B'. Additionally, all transformation matrices are orthonormal.
  • #1
seyma
8
0
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.


Relevant 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.
 
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  • #2
Let's say that the fixed frame is OXYZ and moving frame is OUVW. The transition matrix is R. Then if the rotation is about one of the principal axes of OXYZ (the fixed frame) then the new rotational transformation is A'. Thus, the previous resultant rotation matrix is premultipled by the new rotational transformation, so we have AA'=RA'=R'B. Similarly, if the rotation is about one of the principal axes of OUVW (the moving frame) then the new rotational transformation is B'. Thus, the previous resultant rotation matrix is postmultipled by the new rotational transformation, so we have AR'=RB'=R'B'. This proves that when basic rotations are combined to find composite rotational transformations, if the rotation is about one of the principal axes of OXYZ (the fixed frame) the previous resultant rotation matrix is premultipled 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.
 

Related to Composition of rotational transformations

1. What is meant by "composition of rotational transformations"?

The composition of rotational transformations refers to the process of combining multiple rotations in order to achieve a final rotation. This is commonly used in geometry and physics to describe the transformation of an object's orientation in space.

2. How is the composition of rotational transformations calculated?

The composition of rotational transformations is typically calculated by multiplying the rotation matrices of each individual rotation. This results in a single rotation matrix that represents the final rotation.

3. What is the difference between a clockwise and counterclockwise rotation?

A clockwise rotation is a rotation in the direction that the hands of a clock move, while a counterclockwise rotation is in the opposite direction. This is important to note when composing rotations, as the order in which they are applied can affect the final result.

4. How does the composition of rotational transformations affect the order of operations?

The composition of rotational transformations follows the same order of operations as regular multiplication. This means that the rotations are applied in the order in which they are multiplied, from left to right. Changing the order of the rotations can result in a different final rotation.

5. Can the composition of rotational transformations be applied to 3D objects?

Yes, the composition of rotational transformations can be applied to 3D objects. In this case, the rotations are represented by a 3x3 rotation matrix, and the final result is a combination of rotations in all three dimensions.

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